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

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