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Curvature estimate of steepest descents of 2-dimensional maximal space-like hypersurfaces on space forms
Received date: 13 Dec 2018
Accepted date: 20 Feb 2020
Published date: 15 Feb 2020
Copyright
For the maximal space-like hypersurface defined on 2-dimensional space forms, based on the regularity and the strict convexity of the level sets, the steepest descents are well defined. In this paper, we come to estimate the curvature of its steepest descents by deriving a dierential equality.
Key words: Space forms; steepest descents; maximal space-like hypersurface
Peihe WANG , Jianchun WANG . Curvature estimate of steepest descents of 2-dimensional maximal space-like hypersurfaces on space forms[J]. Frontiers of Mathematics in China, 2020 , 15(1) : 167 -181 . DOI: 10.1007/s11464-020-0826-0
1 |
Bartnik R, Simon L. Spacelike hypersurfaces with prescribed boundary values and mean curvature. Comm Math Phys, 1982, 87: 131–152
|
2 |
Bian B, Guan P, Ma X, Xu L. A constant rank theorem for quasiconcave solutions of fully nonlinear partial differential equations. Indiana Univ Math J, 2011, 60: 101–120
|
3 |
Caffarelli L, Friedman A. Convexity of solutions of semilinear elliptic equations. Duke Math J, 1985, 52(2): 431–456
|
4 |
Caffarelli L, Spruck J. Convexity properties of solutions to some classical variational problems. Comm Partial Differential Equations, 1982, 7: 1337–1379
|
5 |
Cheng S, Yau S. Maximal spacelike hypersurface in the Lorentz-Minkowski spaces. Ann of Math, 1976, 104: 407–419
|
6 |
Huang A, Ma X, Ou Q. Some new harmonic functions on minimal surfaces related to the curvature of its level sets. Preprint
|
7 |
Korevaar N J. Convexity of level sets for solutions to elliptic ring problems. Comm Partial Differential Equations, 1990, 15(4): 541–556
|
8 |
Longinetti M. Convexity of the level lines of harmonic functions. Boll Unione Mat Ital, 1983, 6: 71–75
|
9 |
Longinetti M. On minimal surfaces bounded by two convex curves in parallel planes. J Differential Equations, 1987, 67: 344–358
|
10 |
Ma X, Ou Q, Zhang W. Gaussian curvature estimates for the convex level sets of p-harmonic functions. Comm Pure Appl Math, 2010, 63: 0935–0971
|
11 |
Ma X, Xu L. The convexity of solution of a class Hessian equation in bounded convex domain in ℝ3. J Funct Anal, 2008, 255(7): 1713–1723
|
12 |
Ma X, Zhang W. The concavity of the Gaussian curvature of the convex level sets of p-harmonic functions with respect to the height. Commun Math Stat, 2013, 1(4): 465–489
|
13 |
Ma X, Zhang Y. The convexity and the Gaussian curvature estimates for the level sets of harmonic functions on convex rings in space forms. J Geom Anal, 2014, 24(1): 337–374
|
14 |
Ortel M, Schneider W. Curvature of level curves of harmonic functions. Canad Math Bull, 1983, 26(4): 399–405
|
15 |
Papadimitrakis M. On convexity of level curves of harmonic functions in the hyperbolic plane. Proc Amer Math Soc, 1992, 114(3): 695–698
|
16 |
Talenti G. On functions whose lines of steepest descent bend proportionally to level lines. Ann Sc Norm Super Pisa Cl Sci, 1983, 10(4): 587–605
|
17 |
Wang P. The concavity of the Gaussian curvature of the convex level sets of minimal surfaces with respect to the height. Pacific J Math, 2014, 267(2): 489–509
|
18 |
Wang P, Liu X, Liu Z. The convexity of the level sets of maximal strictly space-like hypersurfaces defined on 2-dimensional space forms. Nonlinear Anal, 2018, 174: 79–103
|
19 |
Wang P, Qiu H, Liu Z. Some geometrical properties of minimal graph on space forms with nonpositive curvature. Houston J Math, 2018, 44(2): 545–570
|
20 |
Wang P, Wang X. The geometric properties of harmonic functions on 2-dimensional Riemannian manifolds. Nonlinear Anal, 2014, 103: 2–8
|
21 |
Wang P, Zhang D. Convexity of level sets of minimal graph on space form with non-negative curvature with nonnegative curvature. J Differential Equations, 2017, 262: 5534–5564
|
22 |
Wang P, Zhang W. Gaussian curvature estimates for the convex level sets of solutions for some nonlinear elliptic partial differential equations. J Partial Differ Equ, 2012, 25: 1–38
|
23 |
Wang P, Zhao L. Some geometrical properties of convex level sets of minimal graph on 2-dimensional Riemannian manifolds. Nonlinear Anal, 2016, 130: 1–17
|
24 |
Wang P, Zhuang J. Convexity of level lines of maximal space-like hypersurface in Minkowski space. Israel J Math, 2018, 226(1): 295{318
|
25 |
Xu L. A microscopic convexity theorem of level sets for solutions to elliptic equations. Calc Var Partial Differential Equations, 2011, 40(1): 51–63
|
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