SOME ASPECTS OF GEOMETRIC FLOWS IN RIEMANNIAN GEOMETRY

SOME ASPECTS OF GEOMETRIC FLOWS IN RIEMANNIAN GEOMETRYSOME ASPECTS OF GEOMETRIC FLOWS IN RIEMANNIAN GEOMETRYSOME ASPECTS OF GEOMETRIC FLOWS IN RIEMANNIAN GEOMETRYSOME ASPECTS OF GEOMETRIC FLOWS IN RIEMANNIAN GEOMETRYSOME ASPECTS OF GEOMETRIC FLOWS IN RIEMANNIAN GEOMETRYLiouville type theorems and gradient estimates for non-linear heat equations along ancient k-super Ricci flow via reduced geometry 30 3.1 Preliminaries and main results.. 1.1 Gradient Ri

Gradient Ricci soltions and isometry groups

The Ricci flow equation is a geometric evolution equation that deforms the metric g of a Riemannian manifold over time by adjusting it in a way proportional to the Ricci curvature Ric:

The Ricci flow is a one-parameter family of metrics defined on a smooth manifold, parameterized by time, that satisfies a specific mathematical equation Introduced by Hamilton in 1982, the Ricci flow plays a crucial role in addressing significant problems in pure mathematics, including Poincaré’s conjecture and Thurston’s geometrization conjecture It has become a central focus in the study of geometric flows and has contributed to theorems such as the Differentiable sphere theorem and the generalized Smale conjecture For recent advancements in Ricci flow theory, readers can refer to R Bamler's survey paper and its references.

In his paper, Hamilton demonstrated that a compact 3-manifold M with strictly positive Ricci curvature can also possess a metric of constant positive curvature, establishing a strong connection to Poincaré’s conjecture and Smith’s conjecture If both conjectures are validated, this result would naturally follow as a corollary However, the Ricci flow equation is only weakly parabolic, often resulting in finite-time singularities, which complicates efforts to prove the Poincaré conjecture This has led mathematicians to explore singularity models to better understand the topological and geometric aspects of Ricci flows, with the Ricci soliton being a crucial model A Ricci soliton is defined as a Riemannian manifold equipped with a smooth vector field that satisfies the Ricci flow equation.

2 L X g = λg, (1.2) where L is the Lie derivative with respect to X and λ ∈ R In particular, if

X = ∇f where f : M → R is a smooth function, then we say that a triple

(M, g, f ) is a gradient Ricci soliton In this case the equation (1.2) becomes

Ric + Hess f = λg, (1.3) where Hess is the Hessian of metric g Depending on the value of λ, a gradient Ricci soliton is called shrinking if λ > 0, steady if λ = 0, or expanding if λ < 0.

An Einstein manifold N is a Riemannian manifold characterized by its Ricci curvature Ric being proportional to its metric g, expressed as Ric = λg, where λ is the Einstein constant These manifolds are significant in differential geometry and theoretical physics, especially in general relativity, as they represent space-times with constant curvature An Einstein manifold exemplifies a gradient Ricci soliton, where the Hessian operator on the potential function f equals zero, making λ the Einstein constant Additionally, the Gaussian soliton serves as another fundamental example.

In the context of R n, g R n, λ|x|^2, we explore cylinders S k × R n−k equipped with the product metric, where the sphere exhibits Ricci curvature λ Additionally, the term "rank k rigid gradient Ricci soliton" is used to describe a combination of the aforementioned concepts, as noted by Petersen and Wylie [81] Notably, this soliton is isometric to a suitable quotient of the underlying structure.

N k × R n−k , with f = |x| 2 2 defined on the Euclidean factor [80] Consequently, a soliton is called non-trivial (or non-rigid) if at least a factor in its de Rham decomposition is non-Einstein.

The study of isometric groups is crucial for classifying the geometric structure of smooth manifolds Dantzig-Waerden’s seminal paper on the isometry group of connected, locally compact metric spaces marked the beginning of significant research in this area Myers and Steenrod demonstrated that the isometry group Iso(M) of a Riemannian manifold M functions as a Lie transformation group under the compact-open topology Subsequently, Kobayashi identified the maximal dimension of Iso(M) and established that the Riemannian manifold M possesses constant curvature.

Iso(M) achieves maximal dimension, and while non-gradient Ricci solitons have been identified in various Lie groups and homogeneous spaces, Petersen and Wylie proved that all homogeneous gradient Ricci solitons exhibit rigidity Additionally, they established that a reducible Riemannian metric corresponds to a reducible soliton structure, relying on splitting results influenced by Killing vector fields.

In Chapter 2, we will explore the isometry group Iso(M) and its corresponding Lie algebra related to an irreducible non-trivial gradient Ricci soliton (M, g, f), drawing inspiration from the work of Petersen and Wylie It is important to note that a Riemannian manifold is classified as irreducible if it cannot be represented as a direct product of lower-dimensional manifolds through any finite cover in an isometric manner.

In this article, we investigate the upper bound on the dimension of the Lie algebra of Killing vector fields associated with irreducible non-trivial gradient Ricci solitons We aim to classify the specific spaces where this maximum dimension is achieved, providing insights into the geometric properties and symmetries of these solitons Through our analysis, we establish key results that enhance the understanding of the structure and behavior of gradient Ricci solitons in differential geometry.

Nonlinear parabolic equations and super geometric flows

The concept of super Ricci flow, initially introduced by McCann and Topping within the realm of optimal transport theory, expands upon the framework of geometric flow theory A smooth manifold (M, g(x, t)) for t ∈ I is defined as a super Ricci flow.

The monotonicity property of the Wasserstein distance along heat flow characterizes supersolutions to the Ricci flow equation, as discovered by the authors in [70] Subsequent research has explored various characterizations through significant geometric inequalities of manifolds, Bakry-Émery gradient estimates, and the convexity of entropies (see [45, 64, 66, 96]) Building on these findings, Sturm [96] introduced the concept of super Ricci flow in time-dependent (non-smooth) metric measure spaces, paving the way for new investigations into super Ricci flow from a metric measure geometry perspective Recently, Bamler [6] showed that the space of super Ricci flows, when appropriately pointed, is compact in a specific topology.

For each k ∈ R, a time-dependent Riemannian manifold (M, g(x, t)) t∈I is termed a k-super Ricci flow if it satisfies the following condition:

The equation ∂t + 2 Ric ≥ 2kg represents a natural extension of the super Ricci flow concept, where the k-super Ricci flow serves as a time-dependent version of a Riemannian manifold with Ricci curvature bounded below by k Notably, the (0)-super Ricci flow is equivalent to the super Ricci flow When equality holds in this equation, the manifold (M, g(x, t)) for t in I is referred to as k-Ricci flow An ancient k-super Ricci flow is characterized by the interval I = (−∞, 0], extending the well-known ancient Ricci flow concept that significantly contributes to the analysis of singularities in Ricci flow.

Perelman introduced the concepts of reduced distance and reduced volume as essential tools for analyzing the Ricci flow Ye later established several properties of these concepts and provided estimates for reduced volume, demonstrating their applications in the analysis of the asymptotic limits of κ-solutions of the Ricci flow Recently, Kunikawa and Sakurai advanced this field by obtaining Liouville type theorems for harmonic maps under ancient super Ricci flow with controlled growth, building on Perelman’s reduced geometric perspective This work continues the exploration of similar topics for functions in previous research.

The next chapter of this thesis is also motivated from a work due to Ma [67] In

[67], for some constants a, b, the author considered the following nonlinear elliptic equation

In a complete noncompact Riemannian manifold, the equation ∆u + au ln u + bu = 0 (1.6) is closely related to the gradient Ricci soliton equation (1.3) as noted by Ma in [67] By taking the trace of equation (1.3), we can derive significant insights into the relationship between these equations.

Here S is the scalar curvature of M and n is the dimension of M According to Proposition 2.3 in Chapter 2, we get

|∇f | 2 + S − 2λf = A 0 , where A 0 is a constant Combining the two above equations, we have

If we set u = e −f , then by a simple computation, it follows that u solves

The equation presented is a specific instance of equation (1.6), which is intrinsically connected to geometric and functional inequalities on manifolds Notably, it relates to the logarithmic Sobolev inequality and Perelman's work in this area.

W-entropy [78] Replacing u by e a b u, we see that the equation (1.6) is equivalent to the following equation

In Chapter 3, we examine gradient estimates for positive bounded solutions to the parabolic version of equation (1.8) under ancient k-super Ricci flow, drawing inspiration from the research of Kunikawa, Sakurai, and Ma Our focus will be on exploring various applications of these gradient estimates.

Problem 1.2 Establish gradient estimates and Liouville type results for positive bounded solutions of the nonlinear parabolic equation related to Perelman’s reduced distance

∂t u(x, t) = ∆u(x, t) + au(x, t) ln u(x, t) (1.9) along ancient k-super Ricci flow, where a ∈ R

A smooth metric measure space, also referred to as a weighted manifold, serves as a natural extension of gradient Ricci solitons Following Perelman's groundbreaking research, this concept has garnered significant attention from mathematicians globally A smooth metric measure space is defined as a triple (M, g, e −f dà), where (M, g) represents a complete Riemannian manifold with a dimension of n ≥ 3, accompanied by a weighted measure e −f dà, where f is a smooth function on M and dà denotes the standard Riemannian volume measure associated with metric g Within this framework, the weighted Laplacian ∆ f is established.

The equation ∆f ã := ∆ã −⟨∇f, ∇ã⟩ represents a natural extension of the Laplace-Beltrami operator ∆ within the framework of smooth metric measure spaces This formulation aligns with the traditional Laplace-Beltrami operator when the potential function f is constant Additionally, for any real number m ≥ 0, the m-Bakry-Émery curvature is defined, further enriching the study of geometric analysis in this context.

Ric m f := Ric + Hessf − 1 m df ⊗ df.

When m = 0, it means that f is constant and Ric m f becomes the usual Ricci curvature Ric When m → ∞, we have the (∞-)Bakry-Émery Ricci curvature

It is evident that if the Ricci curvature \( Ric \, f \) is greater than or equal to a constant \( c \), then \( Ric \, m \) must also satisfy \( Ric \, m \geq c \) However, the reverse implication may not hold true When \( Ric \, f \) is bounded from below, various geometric properties associated with manifolds that have their Ricci tensor similarly bounded can potentially be applied to smooth metric measure spaces; nonetheless, additional assumptions regarding \( f \) are necessary for this extension For more comprehensive insights, refer to sources [63, 103].

Inspired by the works of Hamilton, McCann-Topping, and Perelman on the modified Ricci flow, often referred to as the Perelman-Ricci flow, X.-D Li and colleagues introduced the (k, m)-super Perelman-Ricci flow This concept applies to manifolds with time-dependent metrics and potentials, where k and m are real numbers with m being non-negative A time-dependent smooth metric measure space, denoted as M with metrics g(x, t) and potentials e^(-f(x, t)), is classified as a (k, m)-super Perelman-Ricci flow for t within a specified interval I.

The flow discussed is a weighted variant of the k-super Ricci flow, which is closely related to the curvature-dimension condition CD(k, m) as defined by Sturm and Lott-Villani Notably, as m approaches infinity, the metric g(x, t) and the potential function f(x, t) must adhere to a specific inequality.

∂t + 2Ric f ≥ −2kg, (1.11) we call M, g(x, t), e −f(x,t) dà t∈I a (k, ∞)-super Perelman-Ricci flow, which can be viewed as a natural extended of the modified Ricci flow [78].

The Yamabe problem, a central topic in geometric analysis throughout the 20th century, was introduced by Yamabe in his significant posthumous publication It concerns an n-dimensional smooth, compact Riemannian manifold (M, g) with n ≥ 3 This problem serves as a generalization of the Poincaré-Köbe uniformization theorem, which identifies a constant scalar curvature metric ge that is pointwise conformally related to the original metric g The conformal class of g is defined as [g] = {e^u g : u ∈ C∞(M), u > 0}, highlighting the relationship between different metrics within the same conformal class.

Then the scalar curvature S e g of the conformal metric ge can be written as

The scalar curvature associated with the metric g is denoted as S_g, and the Laplace-Beltrami operator linked to g is represented as ∆ This leads us to understand that the Yamabe problem involves finding a positive solution u to the Yamabe equation.

4(n − 1) S e g u n+2 n−2 = 0 (1.12) where S e g is constant This was resolved through the contributions of N Trudinger

[100], T Aubin [4], and R Schoen [85] Their proofs utilize results from the calculus of variations and elliptic theory; for further details, refer to the survey article by Lee and Parker [60].

The Yamabe flow was initially explored by Hamilton in the unpublished work

[42] as a tool for addressing the Yamabe problem An n-dimensional manifold (M, g(x, t)) t∈I equipped with a time-dependent metric is referred to as a Yamabe flow when it satisfies the following equation

The equation ∂t = −Sg illustrates the relationship between the normalized Yamabe flow and scalar curvatures of the metric g Chow's research demonstrated that this flow converges to a metric with constant scalar curvature Ye further advanced this work by proving the convergence of the Yamabe flow under the assumption of a locally conformally flat initial metric, thereby enhancing Chow's findings Additionally, the behavior of metrics that are not conformally flat has been explored in a series of studies by Schwetlick and Struwe, followed by further investigations from Brendle.

Inspired by the work presented in Chapter 3 and the advancements made in the smooth metric spaces discussed earlier, Chapter 4 will investigate the following problem.

Problem 1.3 Study some analytical aspects of a general type of nonlinear parabolic equation concerning the weighted Laplacian

Translating solitons of the mean curvature flow

The final section of this thesis explores mean curvature flows, which are motivated by geometric applications similar to the Ricci flow in Riemannian manifolds This flow serves as a crucial method for classifying hypersurfaces under specific curvature conditions, deriving isoperimetric inequalities, and creating minimal surfaces Additionally, mean curvature flow plays a vital role in modeling the evolution of interfaces in various multiphase physical scenarios.

The concept discussed in Mullins' influential paper highlights its significance as a gradient-like flow of the area functional, which is particularly relevant for addressing issues related to surface energy.

We now recall the definition of mean curvature flow LetX : M n → R n+m be a smooth immersion of an n-dimensional smooth manifold in Euclidean space R n+m

A smooth one-parameter family X t = X (ã, t) of immersions X t : M ì [0, T ) →

R m+n with corresponding images M t = X t (M ) is called the mean curvature flow for a submanifold M in R m+n if it satisfies the following condition

X (x, 0) = X (x), (1.15) for any (x, t) ∈ M × [0, T ), where H (x, t) is the mean curvature vector of M t at

Studying mean curvature flow involves analyzing singularities, particularly the behavior of the second fundamental form relative to the family M t In compact cases, the second fundamental form can become unbounded in finite time We classify the singularities of mean curvature flow into two categories based on the blow-up rate: Type-I and Type-II singularities The geometry surrounding Type-II singularities presents greater challenges, making their study notably more complex compared to Type-I singularities.

A solution to (1.15) is said to be a translating soliton (or simply a translator) if there exists a constant vector V with unit length in R n+m such that

Translating solitons, represented by the equation H = V ⊥, play a crucial role in the mean curvature flow theory as they emerge as blow-up solutions at type II singularities These unique solutions maintain their shape while moving in a constant direction V, highlighting their significance in the study of geometric flows.

Translating solitons, particularly in the context of hypersurfaces, are limited in number, with notable examples being those that are also minimal hypersurfaces According to the equation M t = M + tV, the vector V must remain tangential to the translator, indicating that these solitons can take the form of Mf × L.

L is a line parallel to V, while Mf represents a minimal hypersurface in L ⊥ The study of translating solitons, as highlighted in sources [48, 51], leads to significant insights and a classification framework Xin's recent work [106] explored various geometric properties of translating solitons, such as volume growth, the generalized maximum principle, and Gauss maps, along with functions related to these maps He also provided integral estimates for the squared norm of the second fundamental form, ultimately establishing a rigidity theorem for translators in higher codimensions of Euclidean space Subsequently, Wang, Xu, and Zhao expanded upon Xin's findings by applying integral curvature pinching conditions to the trace-free second fundamental form (see [101]).

Impera and Rimoldi investigated the topological structure at infinity of translating solitons of the mean curvature flow, utilizing the theory of weighted minimal hypersurfaces They established weighted Sobolev inequalities, demonstrating that an f-stable translator can have at most one end Their research also examined the relationship between L²-weighted harmonic 1-forms, cohomology with compact support, and the index of the translator in relation to the generalized Morse index of a stable operator Following their findings, Kunikawa and Sato concluded that any complete f-stable translating soliton cannot support codimension one cycles, implying that any two-dimensional complete f-stable translator must have a genus of zero.

Inspired by the research results on translating solitons mentioned above, in Chapter 5 of this thesis, we are interested in the following problem.

Problem 1.4 Study of the rigidity properties and connectedness at infinity of complete translating solitons in the Euclidean space via the second fundamental form.

On isometry groups of gradient Ricci solitons 14

Preliminaries

This section is to recall auxiliary results on Killing vector fields, group actions on manifolds, and gradient Ricci solitons The main references are [2, 26, 52, 53,

2.1.1 Killing vector fields and group actions on manifolds

This section provides an overview of the fundamental characteristics of Killing vector fields and their connection to the isometry group, while also revisiting essential concepts concerning group actions on manifolds Key references include standard texts [2, 52, 83].

We begin by providing the definition of Riemannian isometries.

Definition 2.1 Let(M, g M ) and (N, g N ) be Riemannian manifolds An isometry from M to N is a diffeomorphism ϕ : M → N such that ϕ ∗ (g N ) = g M

In other words, ϕ is an isometry if for all p ∈ M and tangent vectors X p , Y p ∈

In this sense, we say that ϕ preserves the metric structure In addition, M and

The set of all isometries of a Riemannian manifold (M, g) onto itself forms a group (indeed a Lie group), which is denoted by Iso(M ) and called the isometry group of M.

Definition 2.2 A vector field X on a Riemannian manifold (M, g) is called a Killing vector field if the Lie derivative with respect to X of the metric g vanishes, i.e., L X g = 0.

The following proposition shows the relationship between Killing vector fields and isometries For a proof, we refer the reader to [83, Proposition 8.1.1].

Proposition 2.1 A vector field X on a Riemannian manifold(M, g) is a Killing vector if and only if the local flows generated by X act by isometries.

Proposition 2.1 establishes that killing vector fields, often referred to as infinitesimal isometries, are derived from the integration of vector fields to achieve isometries, highlighting their significant analytical properties.

Proposition 2.2 [83, Proposition 8.1.4] Let X be a Killing vector field on a Riemannian manifold (M, g) If there exists a point p ∈ M such that X p = 0 and (∇X ) p = 0, then X is identical 0.

Remark 2.1 The set of all Killing vector fields on a Riemannian manifold(M, g) is a Lie algebra, and denoted by iso(M, g) Furthermore, by Theorem 8.1.6 in

If the Levi-Civita connection induced by the Riemannian metric g on the manifold M is complete, then every Killing vector field on M is also complete Consequently, the space of isometries iso(M, g) corresponds to the Lie algebra of the isometry group Iso(M).

In this section, we revisit a crucial result regarding the dimension of the Lie algebra iso(M, g) and the group Iso(M, g), which is essential for our proof of Theorem 2.1 According to Lemma 2.1, a connected Riemannian manifold (M, g) of dimension n has a Lie algebra iso(M, g) with a maximum dimension of 1/2 n(n + 1) If the dimension of iso(M, g) reaches this maximum, it indicates that M is a space of constant curvature Additionally, when the dimension of Iso(M) equals 1/2 n(n + 1), M is isometric to specific geometric structures.

(iii) an n-dimensional real projective space,

(iv) an n-dimensional, simply connected hyperbolic space.

In the rest of this subsection, we recall some basic notions about group actions on manifolds following the book by Alexandrino and Bettiol [2].

Definition 2.3 Let G be a Lie group and M a smooth manifold A smooth map l : G × M → M is called a (left) action of G on M, or a (left) G-action on M, if

(i) l(e, x) = x, for all x ∈ M, where e is the identity element of G;

We often write gã x or just gx in place of the more pedantic notation l(g, x).

A right action r : M ì G → M can be defined analogously and we write x ãg or xg.

Definition 2.4 An action is said to be proper if the associated map G × M 7→

G ì M ∋ (g, x) 7−→ (g ã x, x) ∈ M ì M (2.1) is proper, i.e., if the preimage of any compact subset of M × M under (2.1) is a compact subset of G × M.

According to Proposition 3.62 and Theorem 3.65, actions by closed subgroups of isometries are classified as proper Furthermore, it is established that any proper action can be transformed into an isometric action under a specific Riemannian metric.

Definition 2.5 A Riemannian manifold (M, g) is said to be homogeneous if its isometry group acts transitively, i.e., for each pair of points x, y ∈ M there is a

2.1.2 Some basis results on gradient Ricci solitons

In this section, we will review essential concepts and gather preliminary information about Geometric Riemannian Structures (GRS) Consider a GRS represented as (M, g, f) with a dimension of n ≥ 3; the smooth potential function f: M → R adheres to a specific equation.

The equation Ric + Hess f = λg, where λ is a real number, illustrates the relationship between the Ricci curvature (Ric) of a manifold M, the Hessian of a function f (Hess f), and the scalar curvature (S) of M These quantities are interconnected through specific mathematical equations, as outlined in Proposition 2.1 of reference [39].

Proposition 2.3 For any gradient Ricci soliton (M, g, f ), we have

S + |∇f | 2 − 2λf = C (2.4) for some constant C Here ∆ f denotes the f-Laplacian, ∆ f ã := ∆ ã −⟨∇f, ∇ã⟩.

In [82, Proposition 2.1], Petersen and Wylie proved the following result about a Killing field on a GRS.

Proposition 2.4 If X is a Killing field on a gradient Ricci soliton (M, g, f ), then ∇(Xf ) is parallel Moreover, if λ ̸= 0 and ∇(Xf ) = 0, then also Xf = 0.

Remark 2.2 We emphasize that to prove the above result, Petersen and Wylie used the condition that scalar curvature is bounded However, such an assumption can be omitted since, for λ ̸= 0, the scalar curvature of a GRS is always bounded from either below or above [3, Theorem 8.6]) This is enough for the argument to go through.

Consequently, Petersen and Wylie [82] gave the following splitting result.

Lemma 2.2 [82, Corollary 2.2] If X is a Killing field on a GRS (M, g, f, ) then either ∇(Xf ) = 0 or M locally splits a line isometrically The latter means that around each point p, there is a neighborhood U = V × I, where V is an open neighborhood of a submanifold and T p V ⊥ (∇(Xf )) p and I is an open interval. The Riemannian metric in U is the direct product of the induced metrics on each factor, and (V, g |V , f ) is a GRS.

For (M, g, f ) a shrinking gradient Ricci soliton, upon scaling the metric g by a constant, we can assume that λ = 1 2 Then the equation (2.2) takes the form

By adding a constant to f if necessary and the equation (2.4), we may normalize the soliton such that

According to Chen's findings, any shrinking gradient Ricci soliton satisfies the condition S ≥ 0, which implies that the potential function f must also be non-negative Additionally, research by Haslhofer and Müller indicates that this potential function exhibits quadratic growth at infinity Combining these insights leads us to a significant proposition regarding the behavior of the potential function in this context.

Proposition 2.5 Let(M, g, f) be ann-dimensional complete noncompact shrinking gradient Ricci soliton with (2.5) and (2.6) Then, each regular level set of f is a compact set.

Proof For each regular value c ∈ f (M ) , we consider the level set M c of f Since f is a smooth function and {c} is a closed set, M c = f −1 (c) is also a closed set.

By Lemma 2.1 in [44], there exists a point p ∈ M where f attains its infimum and f satisfies the following quadratic growth estimate

, where r(x) is a distance function from p to x, and a + = max{a, 0} for a ∈ R. This and the fact that f ≥ 0 imply that M c is a bounded set and, therefore, M c is a compact set.

Dimension bound and Rigidity

This section is devoted to the proof of our main results Let(M, g, f) be a GRS of dimension n ≥ 3 Recall iso(M, g) := {X is a smooth tangent vector field on M , L X g = 0}.

:= {X is a smooth tangent vector field on M , L X g = 0 = L X f } (2.7)

Then, we see that isof (M, g, f ) ⊂ iso(M, g) is a vector subspace Towards our goal, we will establish the following lemma concerning iso(M, g).

Lemma 2.3 If X ∈ iso(M, g) and g(X, ∇f ) = Xf is constant then

= Y (L X f ) − (L X g) (Y, ∇f ) (2.8) for any Y ∈ T M Since X is a Killing vector field, (L X g) (Y, ∇f ) = 0 Since

L X f = Xf is a constant, Y (L X f ) = 0 Combining these results yields

We now give the proof of Theorem 2.1.

The proof of Theorem 2.1 begins by defining M c as a level set of the function f, utilizing the induced metric g c, which is derived from g restricted to T M c, where c is a regular value of f According to the level set theorem, (M c, g c) forms a smooth submanifold with a co-dimension of one Let X be an element of isof(M, g, f), and denote the local flow generated by the vector field X as φ X t.

Given that L X g = 0 and L X f = 0, it follows that (φ X t ) ∗ g = g and φ X t ∗ f = f, which implies f ◦ φ X t = f According to Proposition 2.1, the map φ X t : M → M generates local isometries, with φ X t (M c ) ⊆ M c Consequently, φ X t induces a map φe X t ≡ φ X t | M c : M c → M c We will now examine the vector field.

Since φe X t is an isometry on M c , we conclude that

Thus, the map π : iso f (M, g, f) →iso(M c , g c )

M c is well-defined Moreover,π is a linear map Next, we will prove thatπ is injective. Suppose that X | M c ≡ 0,where X ∈ iso f (M, g, f ).Since L X g = 0 and L X f = 0, Lemma 2.3 yields

Letp ∈ M c andY ∈ T M be an arbitrary vector field Then, we haveY = Z + W, where Z ∈ T M c and W ∈ T ⊥ M c Since ∇f is a normal vector field of T M c ,

W = η ∇f, where η is a smooth function Therefore, we get

The last equality follows from X | M c = 0 Furthermore, using (2.10), we compute (∇ ∇f X )| p = (−[X, ∇f ] + ∇ X ∇f )| p = (∇ X ∇f )| p = 0 (2.12)

Since Y ∈ T M is an arbitrary vector field, we conclude that (∇X )| p = 0 Since

X p = 0, by Proposition 2.1, we deduce that X ≡ 0 This shows that the map π is injective From Lemma 2.1 and note that dim M c = n − 1, we obtain dimiso f (M, g, f ) ≤ dimiso(M c , g c ) ≤ 1

We analyze the case where the casedimisof (M, g, f) equals \( \frac{1}{2} (n - 1)n \) According to Lemma 2.1, each regular connected component of f, under the induced metric, exhibits constant curvature, making it homogeneous and complete Consequently, the Lie algebra iso f (M, g, f) generates a global group of isometries on (M, g), with transitive action on each regular level set As a result, S remains constant across these level sets, and, by Proposition 2.3, both |∇f| and |∇f| df are closed and locally exact We define t such that dt = |∇f| df, allowing us to express the metric locally as g = dt² + g_t, where g_t represents a family of metrics on the differentiable manifold corresponding to a regular connected component Let L denote the shape operator, and ν be defined as ∂g_t.

Furthermore, by the constancy of |∇f | on each regular connected component, singular values for f : M 7→ R are isolated By continuity, nearby connected components must be obtained from the same model space ( N n−1 , g N ).

Since g t is homogeneous, so is ν, and it suffices to consider its value at a point.

We recall the evolution of the Ricci tensor, Ric t := Ric(g t ), [26, page 109], for normal coordinates,

∆ L ν ij = ∆ν ij + 2 Rm kijl ν kl − Ric ik ν jk − Ric jk ν ik ,

Rm (νd ) ij = 2 Rm kijl ν kl − Ric ik ν jk − Ric jk ν ik

As ν is homogeneous, all spacial derivatives vanish.

Claim If g N is non-flat then ν is a multiple of g N

The claim is substantiated by the isomorphism of g t to a space form, leading to the conclusion that the Ricci curvature is a scalar multiple of the metric Consequently, when viewed as a linear map on the tangent space, the Ricci curvature acts as a multiple of the identity for each t, and this property extends to its derivative If the Riemann curvature tensor Rm(g N) is non-zero, then Rm(ν) can be expressed as a linear combination of a non-trivial multiple of ν and a scalar multiple of the identity, thus confirming the result.

Thus, if Rm(g N ) ̸= 0 there is a local diffeomorphism ϕ : N × I 7→ U, an open neighborhood in M, such that ϕ ∗ (g) = ϕ ∗ (dt 2 + g t ) = dt 2 + F 2 (t)π ∗ g N The result then follows.

Remark 2.3 The case that g N is flat means each level set is an Euclidean space. Their analysis will be carried out in the Appendix.

Next, we will apply Theorem 2.1 to prove Theorem 2.2.

Proof of Theorem 2.2 Since M is locally irreducible, by Lemma 2.2, ∇(Xf ) ≡ 0 for any Killing vector field X ∈ iso(M, g) We then consider two possible cases.

Case 1: λ ̸= 0 By Prop 2.4, Xf = 0 That is, each Killing vector field automatically preserves f Thus, iso(M, g) ≡ iso f (M, g, f ) and the result then follows from Theorem 2.1.

Case 2: λ = 0 If the scalar curvature S of (M, g, f ) is a constant, then from (2.3), we obtainRic ≡ 0,and hence(M, g, f )is Ricci-flat, which is a contradiction to our non-triviality assumption Thus, S is non-constant, and one observes that it is invariant under isometries Hence iso S (M, g, f) : = {X is a smooth tangent vector field on M , L X g = 0 = L X S}

Repeating the argument as in the proof of Theorem 2.1 we have, for M c a regular level set of S, dimiso(M, g) = dimiso S (M, g, f) ≤ dimiso(M c , g c ) ≤ 1

If equality holds, Lemma 2.1 indicates that every regular connected component of S, under the induced metric, exhibits constant curvature Additionally, the quantity |∇S| remains invariant under the isometric action, and the remainder of the proof follows the same structure as in Theorem 2.1.

The proof of Theorem 2.3 begins with the application of Theorem 2.2, establishing that the dimension of the isometry group, dimiso(M, g), is at most 1/2(n - 1)n, with equality occurring only if every connected component of a regular level set of f is a space form (N^(n-1), g_N) Assuming dimiso(M, g) reaches this maximum, it follows that each regular level set is both homogeneous and complete As a result, iso(M, g) represents the Lie algebra of the isometry group for each regular connected component The proof will then be structured into distinct cases.

Case 1: λ > 0 By Proposition 2.5 each regular connected component is compact Then, by Lemma 2.1, the model space ( N , g N ) must be spherical (round sphere or the real projective space) Then, from Theorem 2.2, the Riemannian metric is a local warped product g = dt 2 + F 2 (t)g N By [12, Theorem 1], the Weyl tensor is vanishing, and (M, g) is locally conformally flat The classification of a GRS with such property for λ > 0 is well-known By [54, Theorem 1], (M, g, f ) must be either the Gaussian shrinking gradient Ricci soliton on R n , the round cylinder shrinker on S n−1 × R, or the round sphere shrinker on S n They are all rigid.

Case 2: λ = 0 If the metric g N is flat, that is (M n , g, f ) (n ≥ 3) is a steady gradient Ricci solition with local Euclidean level sets: g := dt 2 + g t = dt 2 + X i h 2 i (t)dx 2 i , where each function h i is smooth, then from the system (2.19) in Appendix, we get

Note that A ′ = (u 0 − A) A We rewrite the system (2.14) as by the first equation of the above system, we obtain u ′ j u j = u ′ i u i for all j, i This implies that there is a smooth function h such that u

′ j u j = h h ′ for all j Then, we have u j h

Thus u j = a j h for some constant a j From this and (2.14), one finds that

One can notice that dl dh = dl dt dt dh = b l h.

Consequently, bh 2 = l 2 + C for some constant C This and (2.15) lead to l ′ = l 2 + C (2.16)

Now, we consider three possible cases.

Case 1: C = 0 Then the equation (2.16) becomes l ′ = l 2 Using this, we find that l(t) = − 1 t + C 1 , h(t) = ± 1

Case 2: C > 0 Then we set C = D 2 for some constant D and the equation (2.16) becomes l ′ = l 2 + D 2 Thus, we have l(t) = D tan (Dt + D 1 ) , h(t) = ± D

√ b cos (Dt + D 1 ) , and u 0 (t) = l(t) + ah(t) = D tan (Dt + D 1 ) ± aD

√ b cos (Dt + D 1 ) , for some constant D 1

Case 3: C < 0 Then we set C = −D 2 for some constant D and the equation (2.16) becomes l ′ = l 2 − D 2 From this, we get l(t) = D (e 2Dt + D 1 ) e 2Dt − D 1 , h(t) = ± 4D 1 D 2 e 2Dt

√ b (e 2Dt − D 1 ) , and u 0 (t) = l(t) + ah(t) = D (e 2Dt + D 1 ) e 2Dt − D 1 ± 4aD 1 D 2 e 2Dt

√ b (e 2Dt − D 1 ) for some constant D 1 > 0 Then we see that the function u 0 blows up as t approaches a finite time Thus, the metric g is incomplete.

This result shows that the metric g N is non-flat Then by Theorem 2.2 and

[12, Theorem 1], (M, g) is locally conformally flat According to [19, Theorem 2], (M, g, f ) is either the Gaussian soliton or isometric to the Bryant soliton.

Finally, we observe that there is a gap in the dimension.

Corollary 2.1 Let (M n , g, f ),with n ̸= 5, be an irreducible non-trivial GRS and let d := dimiso(M, g) If d < 1 2 (n − 1) n then d ≤ 1

Proof of Corollary 2.1 The proof is by contradiction Suppose that d > 1

The isometry group on the manifold (M, g) generates a Lie algebra consisting of complete Killing vector fields, forming a sub-algebra of iso(M, g) The proofs of Theorems 2.1 and 2.2 establish an injective mapping from iso(M, g) to a co-dimension one regular submanifold (M c, g c), while ensuring that the completeness of vector fields is maintained through this mapping Consequently, for each regular connected component, the dimension of Iso(M c, g c) is greater than one.

By [53, Theorem 3.2], dim(Iso(M c , g c )) = 1 2 n(n − 1) and each (M c , g c ) is a space form which is homogeneous and complete Thus, by continuity, we go back to the case of Theorem 2.2 and d = 1 2 n(n − 1), a contradiction.

Liouville type theorems and gradient estimates for non-

On isometry groups of gradient Ricci solitons

This chapter is written based on the paper “Ha Tuan Dung, Hung Tran (2025),

The article titled "On Isometry Groups of Gradient Ricci Solitons," set to be published in Forum Mathematicum, delves into Problem 1.1 from Chapter 1, focusing on the isometry group and its Lie algebra of an irreducible, non-trivial gradient Ricci soliton (M, g, f) The primary objective is to ascertain the maximum dimension of the isometry group and analyze the manifold's structure when this maximum dimension is achieved To facilitate this, we revisit the Lie algebra of the isometry group defined as iso(M, g) := {X is a smooth tangent vector field on M, L_X g = 0}.

Closely related to the Lie algebra iso(M, g) is the Lie algebra of Killing vector fields preserving f: iso f (M, g, f ) := {X is a smooth tangent vector field on M , L X g = 0 = L X f }.

Throughout this chapter, for convenience in presentation, we will abbreviate the term gradient Ricci soliton as GRS.

In order to achieve the main goal, we first give a result estimating the dimension of iso f (M, g, f ) and classify the spaces where this maximal dimension is achieved.

Theorem 2.1 states that for a GRS (M n, g, f) with n ≥ 3, if the function f is non-constant, then the dimension of iso f (M, g, f) is at most 1/2(n - 1)n, achieving equality only when each connected component of a regular level set of f exhibits constant curvature Furthermore, if we denote (N n−1, g N) as the space form model and g N is non-flat, equality is attained if the metric is locally a warped product This means there exists an open dense subset where each point has a neighborhood diffeomorphic to a product I × N, with the metric expressed as g = dt² + F²(t)g N, where I is an open interval and F: I → R⁺ is a smooth function.

Furthermore, it is possible to relax the assumption on preservingf A Rieman- nian manifold is locally irreducible if it is not a local Riemannian product metric around each point.

Theorem 2.2 Let (M n , g, f ), with n ≥ 3, be a locally irreducible non-trivial

GRS Then iso(M, g) is of dimension at most 1 2 (n − 1) n In addition, equality happens iff it is smoothly constructed, as in the case of equality of Theorem 2.1.

The theorems discussed are fundamentally local in nature, lacking any reference to the completeness of the metric The rigidity of the soliton structure poses significant challenges in completing the aforementioned metrics.

Theorem 2.3 Let (M n , g, f ), with n ≥ 3, be an irreducible non-trivial complete

GRS Then iso(M, g) is of dimension at most 1 2 (n − 1) n For λ ≥ 0, equality happens iff λ = 0 and it is isometric to a Bryant soliton.

Chapter 2 is structured into three sections: Section 2.1 reviews essential notations and gathers preliminary materials for the chapter, Section 2.2 presents the main results and their proofs, and the Appendix addresses the scenario where each level set of a GRS is Euclidean.

This section is to recall auxiliary results on Killing vector fields, group actions on manifolds, and gradient Ricci solitons The main references are [2, 26, 52, 53,

2.1.1 Killing vector fields and group actions on manifolds

This section provides a concise overview of the fundamental characteristics of Killing vector fields and their connection to the isometry group, while also revisiting essential concepts regarding group actions on manifolds For further reading, standard references include [2, 52, 83].