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
Bn
RBall with center Oand radius Rin Rn
Gnn-dimensional Gauss space
KGaussian curvature
H, ~
HMean curvature, mean curvature vector
Hf,~
HfMean curvature and mean curvature vector with density
n,NUnit normal vector
Sn−1
RHypersphere with center Oand radius Rin Rn
CRn-dimensional cylinder in Rn+1
L(C)Riemannian length of the curve C
Lf(C)Length of the curve Cwith density e−f
ds, dA Riemannian area element
dsf, dAfArea element with density e−f
dV Riemannian volume element
dVfVolume element with density e−f
r(x)r(x) = px2
1+··· +x2
n, 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
TpΣTangent space of Σat p
δij Kronecker Symbol
∆f;∇fLaplacian and Gradient of the function f
∇XYCovariate derivative of the vector field Yalong X
α(t)Curve α
∂ΩBoundary of region Ω
|x|Norm of vector x
p. i i-th page in the citation
End of proof
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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 R3
142
3.2.5 A part of slice and graph have the same boundary 47
3.2.6 Slice P, entire graph Σand Gnin R+×wGn49
3.2.7 Entire graph Σand Gnin G+×aGn51
3.2.8 Slice P and entire graph Σin G+×aGn52
3.3.9 f-maximal entire graph Σin Gn×R157
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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 Σand the weighted
volume of a region Eare defined as follows
Areaf(Σ) = ZΣ
e−fdA and Volf(E) = ZE
e−fdV,
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−fdV )a Riemannian manifold (M, g)with density e−f.In
particular, if Mis Euclidean space Rnwith dot product and density
e−f,we simply denote (Rn, e−f).
On a weighted manifold (M, g, e−fdV ),M. Gromov (see [26]) ex-
panded the notion of mean curvature Hto weighted mean curvature
of a hypersurface, denote by Hf,is defined by
Hf:= H+1
n−1h∇f, Ni,
where Nis 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
the Gaussian probability plane G2,R2with density 1
2πe−r2/2, is fre-
quently used in statistics and probability.