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. | 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 Catenoid minimal surface 12 Helicoid minimal surface 13 Scherk minimal surface 14 Grim Reaper curve
