Summary of doctoral thesis in mathematics: Some results on f- minimal surfaces in product spaces

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In this thesis, we study some results of f-minimal surfaces in product spaces with the following purposes: State the relation between the f-minimal surfaces and the selfsimilar solutions of the mean curvature flow; state some properties of the f-minimal surfaces in the product spaces; construct some Bernstein type theorems, halfspace type theorems for f-minimal (f-maximal) surfaces in product spaces; state some results on the higher codimensional f-minimal surfaces.

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MINISTRY OF EDUCATION AND TRAINING HO CHI MINH CITY UNIVERSITY OF EDUCATION - - - - - - ∗∗∗ - - - - - - NGUYEN THI MY DUYEN SOME RESULTS ON f -MINIMAL SURFACES IN PRODUCT SPACES Major: Geometry and Topology Code: 62 46 01 05 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS Ho Chi Minh - 2021
This research project is completed at Ho Chi Minh City Univer- sity of Education. Scientific advisors: 1. Assoc. Prof. Dr. DOAN THE HIEU 2. Dr. NGUYEN HA THANH Reviewer 1: Assoc. Prof. Dr. Kieu Phuong Chi Reviewer 2: Assoc. Prof. Dr. Le Anh Vu Reviewer 3: Dr. Nguyen Duy Binh The thesis will be defended under the assessment of Ho Chi Minh City University of Education Doctoral Assessment Committee At . . . . . . hour . . . . . . date . . . . . . month . . . . . . year 2021 This thesis can be found at the library: - National Library of Vietnam - Library of Ho Chi Minh City University of Education - Ho Chi Minh General Science Library
i LIST OF SYMBOLS Symbols Meaning BRn Ball with center O and radius R in Rn Gn n-dimensional Gauss space K Gaussian curvature H, H~ Mean curvature, mean curvature vector ~f Hf , H Mean curvature and mean curvature vector with density n, N Unit normal vector n−1 SR Hypersphere with center O and radius R in Rn CR n-dimensional cylinder in Rn+1 L(C) Riemannian length of the curve C Lf (C) Length of the curve C with density e−f ds, dA Riemannian area element dsf , dAf Area element with density e−f dV Riemannian volume element dVf Volumep element with density e−f r(x) r(x) = x21 + · · · + x2n , with x = (x1 , . . . , xn ) ∈ Rn Area(M ) Area of M Areaf (M ) f -area of M Vol(M ) Volume of M Volf (M ) f -volume of M Tp Σ Tangent space of Σ at p δij Kronecker Symbol ∆f ; ∇f Laplacian and Gradient of the function f ∇X Y Covariate derivative of the vector field Y along X α(t) Curve α ∂Ω Boundary of region Ω |x| Norm of vector x p. i i-th page in the citation End of proof
ii LIST OF FIGURES Figure Figure’s name Page 1.2.1 Catenoid minimal surface 12 1.2.2 Helicoid minimal surface 13 1.2.3 Scherk minimal surface 14 1.4.4 Grim Reaper curve 20 2.1.1 Density of Gauss space is concentrated in the origin 22 3.1.2 Cylinder is a warped product space 38 3.1.3 Hyperboloid of one sheet is a warped product space 38 3.1.4 Catenoid is a warped product space 38 3.1.5 Spacelike, timelike and lightlike vectors in R31 42 3.2.5 A part of slice and graph have the same boundary 47 3.2.6 Slice P, entire graph Σ and Gn in R+ ×w Gn 49 3.2.7 Entire graph Σ and Gn in G+ ×a Gn 51 3.2.8 Slice P and entire graph Σ in G+ ×a Gn 52 3.3.9 f -maximal entire graph Σ in Gn × R1 57
1 INTRODUCTION A weighted manifold (also called a manifold with density) is a Riemannian manifold endowed with a positive, smooth function e−f , called the density, used to weight both volume and perimeter elements. The weighted area of a hypersurface Σ and the weighted volume of a region E are defined as follows Z Z −f Areaf (Σ) = e dA and Volf (E) = e−f dV, Σ E where dA and dV are the Riemannian area and Riemannian volume elements, respectively. In terms of symbols, we often denote by triple (M, g, e−f dV ) a Riemannian manifold (M, g) with density e−f . In particular, if M is Euclidean space Rn with dot product and density e−f , we simply denote (Rn , e−f ). On a weighted manifold (M, g, e−f dV ), M. Gromov (see [26]) ex- panded the notion of mean curvature H to weighted mean curvature of a hypersurface, denote by Hf , is defined by 1 Hf := H + h∇f, Ni, n−1 where N is the unit normal vector field of the hypersurface. The above definition has been tested to satisfy the first and second vari- ations of the weighted area function (see [40]). The notions of volume, perimeter, curvature, mean curvature, minimal surface,... with density are also simply called f -volume, f - perimeter, f -curvature, f -mean curvature, f -minimal surface,... Weighted manifold relative to physics. In physics, an object may have differing internal densities so in order to determine the object’s mass it is necessary to integrate volume weighted with density. In addition, weighted manifold is also related to the economy when 1 −r2 /2 the Gaussian probability plane G2 , R2 with density 2π e , is fre- quently used in statistics and probability.
2 Weighted manifolds appeared quite a while in mathematics un- der another names “mm-spaces”. Later, Professor Morgan called this class of manifolds “manifolds with density” (see [40]). Recently, the weighted manifold is a new area of interest to be studied by many mathematicians, including Professor Morgan and his team. They proved the solution of the weighted isometric prob- lem, if it exists, its boundary must have f -constant mean curvature (see [14]). A hypersurface Σ in (M, g, e−f dV ) (resp. in (M, g)) is called f -minimal or f -maximal (minimal or maximal) if f -mean curva- ture (mean curvature) of Σ satisfies Hf (Σ) = 0 (H(Σ) = 0). If Hf (Σ) = λ is a constant then Σ is called a λ-hypersurface. The problem of researching the theory, investigating the properties of f -minimal surfaces and constant f -mean curvature surfaces in the weighted manifolds has been receiving a lot of attention from math- ematicians. The authors C. Ivan, H. Neil, H. Stephanie, Ă. Vojislav and Y. Xu showed some surfaces with constant f -mean curvature in Gauss space, investigated some geometrical properties of the sur- faces with constant f -mean curvature (see [14]). J. M. Espinar and H. Rosenberg investigated some geometrical properties of the com- plete surfaces with constant f -mean curvature (see [13]). D. T. Hieu and N. M. Hoang classified f -minimal cylindrical ruled surfaces in R3 with log-linear density (see [31]). On the other hand, the f -area minimizing property of f -minimal hypersurfaces is also of interest. For example, D. T. Hieu applied the weighted calibration method to prove that some submanifolds were f -area minimizing (see [29]). The f -minimal surface in the Gauss space is shrinker, a self- similar solution of mean curvature flow, plays an important role in studying the singularities of the mean curvature flow (see [12]). This is also a matter of interest to research today: mean curvature flow, self-similar solutions of the mean curvature flow, their relationships to f -minimal hypersurfaces in the weighted spaces (see [24], [47],
3 [48]). In recent years, the minimal surfaces in product spaces have been studied by Harold Rosenberg and his partners (see [13], [44]). It is a topic attracting the interest of many mathematicians. Note that the Gauss space is also a space with the product density Gn = G1 × . . . × G1 . In the direction of extending the classical theorems of differential geometry to the weighted spaces and manifolds, many results have been published such as Gauss-Bonnet theorem (see [15]), Liouville’s theorem for bounded harmonic functions in the weighted spaces (see [36]),... However, some classical theorems do not hold when adding density. For example, the four-vertex theorem is not true on the plane with spherical density (see [33]). Accordingly, extending the results on the classical Bernstein theorem, the classical halfspace theorem to obtain the Bernstein-type theorems, the halfspace-type theorems with the extensions to the higher codimension surfaces, to the product manifolds (Riemann product, warped product, Lorentz product,...) or to the weighted manifolds,... are current issues being studied by many authors (see [1], [24], [28], [32], [44], [47], [48]). Deriving from the necessary to find out and solve the above problems, we choose a research topic for this thesis as “Some results on f -minimal surfaces in product spaces”. Here, the thesis mentions two important theorems, related to the main results of the thesis: Bernstein’s theorem and the halfspace theorem. The classical Bernstein’s theorem and its expansions The classical Bernstein’s theorem asserts that an entire minimal graph over R2 is a plane in R3 (see [43]). This result has been proved by Bernstein in the years 1915-1917. Many mathematicians tried to generalize the Bernstein’s theorem to higher dimensions as well as higher codimensions. In 1965, De Giorgi proved Bernstein’s theorem
4 for entire minimal graphs over R3 in R4 (see [17]). In 1966, Almgren proved the theorem in R5 (see [2]). In 1968, Simons extended the theorem to R8 . He proved that an entire minimal graph of dimension n has to be planar for n ≤ 7 (see [46]). In 1969, Bombieri, De Giorgi, and Giusti produced a counterexample in dimension 8 and higher (see [5]), this proves that the result of Bernstein’s theorem only holds for n ≤ 7. Thus, the proof of Bernstein’s theorem for minimal hypersurfaces in Rn is considered to be solved. In the theory of minimal hypersurfaces, Bernstein’s theorem is one of the most fundamental theorems. Thus, it is natural to ask whether there is a Bernstein type theorem in an ambient space other than Rn , such as Riemannian manifolds, Lorentz-Minkowski spaces, warped product spaces, manifolds with density,... Similarly, Bernstein’s theorem is also stated for maximal hy- persurfaces in Lorentz-Minkowski space Rn+1 1 . Different from Bern- stein’s theorem for minimal surfaces in R n+1 , it holds true for all n for maximal surfaces in Lorentz-Minkowski space Rn+1 1 (see [8]). Mathematicians have been expanding Bernstein’s theorem to ob- tain Bernstein type theorems in many different ways. The classical halfspace theorem and its expansions Another theorem related to the minimal surfaces that has at- tracted the interest of mathematicians in recent years is the half- space theorem. The classical halfspace theorem by Hoffman and Meeks (see [35]) states that two complete properly immersed mini- mal surfaces in R3 must intersect unless they are parallel planes. When we replace one of the two surfaces above by a plane, we get the weak halfspace theorem, also called the halfspace theorem. It has been shown that these halfspace theorems do not hold true in the case of the higher dimensions. Therefore, mathematicians are concentrating on extending the halfspace theorem in different ways so that we can be obtained halfspace type theorems, such as:
5 - Extending to the weighted manifolds; - Extending to product spaces; - Extending to class of f -minimal surfaces with one and higher codimensions,... • In this thesis, we study some results of f -minimal surfaces in product spaces with the following purposes: - State the relation between the f -minimal surfaces and the self- similar solutions of the mean curvature flow. - State some properties of the f -minimal surfaces in the product spaces. - Construct some Bernstein type theorems, halfspace type theo- rems for f -minimal (f -maximal) surfaces in product spaces. - State some results on the higher codimensional f -minimal sur- faces. • To achieve our goals, we do the following steps: - Choose some certain product spaces (Riemannian product, warped product, Lorentzian product). - On the selected product space, consider f -minimal surfaces (with one or higher codimension), then explore and state: some properties, Bernstein type theorems, halfspace type theorems for the f -minimal surfaces. • Technically, we use the following main research methods: - Differential and Integral Calculus in calculations. - Variable method to determine the variation of weighted area formula. - In particular, calibration combine with Stokes’ theorem to prove the area-minimizing properties is used throughout Chapter 3 of this thesis. - The general divergence’s theorem to construct the f -volume formula of m-shrinkers and the results related to the higher codi- mension Bernstein type theorems in Gn . With the title and research purposes as above, the main content
6 of this thesis is presented in three chapters: Chapter 1: “A brief overview of the minimal surfaces”. In this chapter, synthesized from references, the thesis presents a summary of some notions, properties, examples, and important results related to the minimal surfaces. Chapter 2: “f -minimal surfaces”. In this chapter, the sections 2.1, 2.2 and 2.3 are synthesized from some references to present an overview of the weighted manifolds and some notions, properties, important results on f -minimal surfaces. In particular, in section 2.4 about the mean curvature flow and the self-similar solutions of the mean curvature flow, we have obtained some main results of this thesis. Chapter 3: “Some results on f -minimal surfaces in product spaces”. In order to make it easier for the readers to follow this chapter, the thesis dedicated to the section 3.1 to introduce some notions and properties of product spaces (Riemann product, warped product, Lorentz product) and weighted product spaces, they are synthe- sized from several references. Then, in sections 3.2, 3.3 and 3.4 the thesis presents some main results that we achieve on f -minimal (f - maximal) surfaces in certain product spaces. In addition, the thesis has presented the corresponding summary and conclusion at the beginning and the end of each chapter to make it easier for readers to grasp the main contents and results of that chapter.
7 Chapter 1 A BRIEF OVERVIEW OF THE MINIMAL SURFACES In this chapter, the thesis presents a summary of the minimal surfaces by synthesized from references [11], [12], [16], [18], [27], [38], [42], [49], [50], including: some basic notions such as mean curvature and mean curvature vector of the hypersurfaces in Rn , vector mean curvature of a k-dimensional submanifold in an n-dimensional Rie- mannian manifold; some examples and important results on the minimal surfaces; and finally the notion of mean curvature flow, self-similar solutions of the mean curva- ture flow (shrinkers or translators). 1.1 Mean curvature 1.1.1 Mean curvature of hypersurfaces in Rn Definition 1.1.1.3. Let Σ be a hypersurface in Rn and a point p ∈ Σ. Then 1. The mean curvature of the hypersurface Σ at p, denoted by H(p), is defined 1 H(p) = tr Sp . n−1 Therefore, we have n−1 1 X H(p) = κi (p), n−1 i=1 where κi (p) are the principal curvatures at p of Σ. ~ := HN is called mean curvature vector of Σ. 2. Vector H 3. Determinant of Sp is called Gauss curvature at p of Σ, denoted by K(p).
8 Consider the graph hypersurface Σu of the smooth function u : U ⊆ Rn−1 −→ R, parameterized by r(x) = x, u(x) . Then, the unit normal vector field of Σu is calculated by 1 N= p (−∇u, 1). 1 + |∇u|2 Therefore, the mean curvature of Σu is given by ! 1 ∇u H= div p . n−1 1 + |∇u|2 1.1.2 Mean curvature of submanifolds Definition 1.1.2.3. 1. Second fundamental form of Σ is a symmetric bilinear form A, defined by A(X, Y ) = (∇X Y )N . ~ of Σ at p is defined by 2. The mean curvature vector H k ~ = 1 X H A(ei , ei ), k i=1 where {ei } is an orthonormal basis of Tp Σ. If Σ is a hypersur- face with the unit normal vector N such that e1 , . . . , en−1 , N ~ is positively oriented then H := H.N is called mean curvature of Σ at p. 3. Let X be a vector field on Σ, then divergence of X at p ∈ Σ, denoted by divΣ X, is defined by Xk divΣ X = gp (∇ei X, ei ). i=1 Remark 1.1.2.4. If Σ is a hypersurface then we have n−1 X n−1 X (n − 1)H = g A(ei , ei ), N = − g (∇ei N, ei ) = − divΣ N, i=1 i=1 where N is the unit normal vector of Σ at p.
9 1.2 Minimal surfaces. Examples 1.2.1 Minimal surfaces Definition 1.2.1.1. A surface Σ is called minimal if its mean cur- vature is equal to zero at all points, i.e., H ≡ 0. In Rn , consider the minimal graph Σ, defined by xn = f (x1 , · · · , xn−1 ), parameterized by r(x1 , . . . , xn−1 ) = (x1 , . . . , xn−1 , f (x1 , . . . , xn−1 )). From the condition H = 0, it follows that i,j g ij fij = 0. P In particular, when n = 3, we get the minimal hypersurface equation (Lagrange equation): (1 + fy2 )fxx − 2fx fy fxy + (1 + fx2 )fyy = 0, where we have denoted x = x1 , y = x2 . 1.2.2 Examples Cateniod; Helicoid; Scherk surface. 1.3 Some important results on minimal surfaces - The first variation formula shows that the minimal surface is the one whose mean curvature is equal to zero at all points. Equiv- alently, it is the extreme points of the area function. - The following proposition gives us another important result (see [50]). Proposition 1.3.5. The minimal graph in Rn is area-minimizing in its homology class. - The following result is related to the Laplace operator (see [50]). Corollary 1.3.7. An isometric immersion ψ : M −→ Rn is a minimal immersion if and only if each component of ψ is a harmonic function of M. 1.4 Mean curvature flow. Self-similar solutions Definition 1.4.1. Mean curvature flow is the family (one param- eter) (Ft )t∈I of smooth immersions moving in the normal vector
10 direction with a velocity equal to the mean curvature of the sur- face. The 1-dimensional case of the mean curvature flow is called curve shortening flow. More specifically: Let M be a hypersurface in Rn+1 . A time-dependent immersion Ft = F (., t) : M −→ Rn+1 , where t ∈ [0, T ] ⊂ R, is a solution of the mean curvature flow if ∂ F (p, t) = H(p, t).N(p, t), p ∈ M, t ∈ [0, T ], ∂t with H(p, t), N(p, t) are the mean curvature and unit normal vector of the hypersurface Ft (M ) at Ft (p), respectively. In the solutions of the mean curvature flow, there is a special solution called self-similar. We consider two types of self-similar solutions that are self-shrinkers (or shrinkers) and translators. 1.5 Conclusions of Chapter 1 In Chapter 1 of this thesis, we briefly introduced the minimal surface from a number of references, including: - Some notions related to regular, smooth parametric hypersur- faces in Rn . - Some necessary basic notions such as mean curvature, mean curvature vector of a hypersurface in Rn and of a k-dimensional submanifold in an n-dimensional Riemannian manifold; minimal surface; mean curvature flow, similar solutions of mean curvature flow (shrinkers or translators). - Some important results on the minimal surfaces such as: + The minimal surfaces are the extreme points of the area function. + The minimal graph is area-minimizing in its homology class. + An isometric immersion ψ : M → Rn is minimal if and only if each component of ψ is a harmonic function on M.
11 Chapter 2 f -MINIMAL SURFACE In this chapter, from references [14], [20], [26], [29], [31], [32], [40] the thesis presents an overview of the weighted manifolds and some common specific densities such as Gaussian density, radial density, log-linear density, etc. Then we have the notion of f -minimal surface and ex- amples of the f -minimal surfaces. In addition, some im- portant results on f -minimal surfaces are also introduced in this chapter. Finally, in the Section 2.4, the thesis builds and proves the relations between the f -minimal surfaces and the self-similar solutions of the mean curvature flow, these results are extracted from [20] in the list of author’s articles related to the thesis. 2.1 Weighted manifold Definition 2.1.1. Weighted manifold or manifold with density is an n-dimensional smooth Riemannian manifold with a smooth positive density e−f used to weight the k-dimensional volume. Let dV be a k- dimensional Riemannian volume element. Then, the k-dimensional volume element with density e−f , denoted by dVf , is defined by dVf = e−f dV. Definition 2.1.4. Gauss space Gn is Rn with Gaussian density n r2 (2π)− 2 e− 2 , where r is the distance from the point to the origin. In particular, the Gauss plane is the Euclidean plane R2 with density r2 (2π)−1 e− 2 . The generalization of the notion of Gauss space is the notion of manifold with radial density, the density function e−f (r) , where r is the distance from the point to the origin, f is a smooth function.
12 In addition, another common weighted space is space with log- linear density, Rn with density e−f (x) , where f (x) = ni=1 ai xi + b, P ai , b ∈ R, ni=1 a2i 6= 0. P Definition 2.1.5. Let ω be a differential k-form on manifold M with density e−f . Then, weighted external differential or f -external differential of ω is defined by df ω := ef d(e−f ω). The differential form ω is called df -closed if df ω = 0, is called df - exact if there exists a differential form η such that ω = df η. In this thesis, the calibration method is used a lot to prove the main results. Stokes’ theorem is the main tool for proving the prin- ciple of calibration. In the weighted spaces, we have: Proposition 2.1.6. Let ω be a differential form on the weighted oriented manifold M. Then we have Z Z −f e df ω = e−f ω. M ∂M 2.2 f -minimal surfaces. Examples Definition 2.2.1. On a Riemannian manifold M n with density e−f , weighted mean curvature or f -mean curvature, Hf of a hypersurface Σ with the outward unit normal vector N is given by 1 Hf = H + h∇f, Ni , n−1 where H is the Riemannian mean curvature of Σ. Note that in some documents, the mean curvature is determined by H(p) = tr Sp , then the f -mean curvature is Hf = H + h∇f, Ni . Definition 2.2.1. On a Riemannian manifold M with density e−f , the hypersurface Σ is called weighted minimal or f -minimal if its f -mean curvature is equal to zero at every points on the surface.
13 2.3 Some important results on f -minimal surfaces - The f -mean curvature of the hypersurface also satisfies the first variation formula with density of the f -area function. Thus, a f -minimal hypersurface is equivalent to being an extreme of the f -area function. - We have the principle of calibration as follows: Proposition 2.3.2. Every weighted calibrated submanifold with or without boundary is weighted area-minimizing in its homology class. - The following theorem gives us another important result: Theorem 2.3.3. Let Σ be the graph of a twice differentiable func- tion u : U ⊂ Rn−1 −→ R. If Σ is a f -minimal hypersurface in Rn = Rn−1 × R with density e−f (x1 ,...,xn−1 ) then Σ is local f -area minmizing . Definition 2.3.2. Let Σ be a hypersurface in Rn with density e−f and X : U ⊂ Rn−1 −→ Rn is a parameterization of Σ. Then, f - Laplace of X is defined by ∆f X := ∆X + h∇f, NiN. - We get an important result as in Rn as follows: Theorem 2.3.9. Let X : U ⊂ Rn−1 −→ Rn be an orthogonal parameterization of the hypersurface Σ, then X is f -minimal if and only if ∆f X = 0. The following are some main results of this thesis on the relation between the f -minimal surface and the similar solutions of the mean curvature flow. These results are extracted from [20] in the list of author’s articles related to the thesis. 2.4 Relation between the f -minimal surfaces and the self- similar solutions of the mean curvature flow 2.4.1 f -minimal surfaces and shrinkers Proposition 2.4.1.1. The self-shrinking solution of the mean cur- n vature flow in Rn , shrinker is a f -minimal hypersurface in G , Rn
14 2 with density e−r /4 . 2.4.2 f -minimal surfaces and translators Proposition 2.4.2.1. The translating solution of the mean curva- ture flow, translator is a minimal surface with the log-linear density. 2.5 Conclusions of Chapter 2 In Chapter 2 of this thesis, we have solved the following issues: - Present an overview of the weighted manifolds, examples and some notions. Introduce some common specific densities such as Gaussian density, radial density, log-linear density. - A brief introduction to the f -minimal surfaces, examples, and some important results on f -minimal surfaces such as: + A f -minimal hypersurface is equivalent to being an extreme of the f -area function. + Every weighted calibrated submanifold with or without bound- ary is weighted area-minimizing in its homology class. + If Σ is a f -minimal hypersurface in Rn = Rn−1 × R with density e−f (x1 ,...,xn−1 ) then Σ is local f -area minimizing. + From the notion of f -Laplace operator, we obtain that an orthogonal parameterized hypersurface X is f -minimal if and only if ∆f X = 0. - In particular, we have established and proved the relation be- tween the f -minimal surface and the self-similar solutions of the mean curvature flow. We have proven: + The self-shrinking solution of the mean curvature flow in Rn , n 2 shrinker is a f -minimal hypersurface in G , Rn with density e−r /4 . (Proposition 2.4.1.1). + The translating solution of the mean curvature flow, the translator is a minimal surface with the log-linear density (Proposi- tion 2.4.2.1).
15 Chapter 3 SOME RESULTS ON f -MINIMAL SURFACES IN PRODUCT SPACES In this chapter, the thesis presents its main results on f -minimal surfaces in the weighted product spaces, where the product can be Riemannian product, warped product or Lorentzian product. For the convenience of the reader, the thesis dedicates the first section 3.1 to briefly present some notions of weighted product mani- folds or product of weighted manifolds, warped product, Lorentzian product and some basic properties in these spaces. They are extracted from references [39], [42], [45]. Then, by considering each product space with a specific density, in the next sections 3.2, 3.3 and 3.4, the thesis ob- tained some results on f -minimal (f -maximal) surfaces in weighted product spaces including: some (local or global) weighted area-minimizing properties of slices in certain weighted warped products; some Bernstein type theorems for f -minimal (f -maximal) hypersurfaces; finally, some re- sults related to the Bernstein type theorems and halfspace type results for the f -minimal surfaces both codimension 1 and higher codimensions in Gauss space. The obtained results in this chapter are extracted from articles [3], [21], [22], [30] in the list of author’s articles related to the the- sis. 3.1 The weighted spaces with Riemannian product, warped product, Lorentzian product Definition 3.1.5. Let M1 and M2 be two weighted manifolds, where the density functions are e−h1 (x) and e−h2 (y) , respectively. On the product manifold M1 × M2 , we consider the density function
16 e−f (x,y) = e−(h1 (x)+h2 (y)) with x ∈ M1 , y ∈ M2 . Then, the weighted k-dimensional volume of M1 × M2 is defined by dVf = e−(h1 +h2 ) dVM1 dVM2 . Then M1 × M2 with such density is called weighted product manifold (product manifold with density) or product of weighted manifolds. Notice that with the above definition, we can consider Gauss space to be the product of Gauss lines Gn = G1 × · · · × G1 (n multipliers). Moreover, according to Proposition 2.4.1.1, shrinker is a f -minimal surface in Gauss space, a weighted product space. This is the rea- son why in some main results of this thesis we consider shrinkers while the title of the thesis is “Some results on f -minimal surfaces in product spaces”. In the following section 3.2, we present some main results on f - minimal surfaces in the weighted warped product R+ ×w Gn . These results are extracted from [21] in the list of author’s articles related to the thesis. 3.2 Some results on f -minimal surfaces in warped product R+ ×w Gn A theme widely approached in recent years is problems con- cerning to hypersurfaces in a warped product manifold of the type R+ ×w M, where R+ = [0, +∞), (M, g) is an n-dimensional Rieman- nian manifold and w is a positive smooth function defined on R+ . Note that with these ingredients, the product manifold R+ ×w M is endowed with the Riemannian metric g¯ = πR∗ + (dt2 ) + w(πR+ )2 σM ∗ (g), where πR+ and πM denote the projections onto R+ and M, respec- tively.