GRADIENT ESTIMATES FOR POSITIVE SOLUTIONS OF THE HEAT EQUATION UNDER GEOMETRIC FLOW

被引：47

作者：

Sun, Jun ^{[1
]}

机构：

[1] Chinese Acad Sci, Inst Math, Acad Math & Syst Sci, Beijing 100190, Peoples R China

来源：

PACIFIC JOURNAL OF MATHEMATICS | 2011年 / 253卷 / 02期

关键词：

gradient estimate; geometric flow; heat equation; Harnack inequality; MEAN-CURVATURE; HARNACK ESTIMATE; KERNEL;

D O I：

10.2140/pjm.2011.253.489

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We establish first- and second-order gradient estimates for positive solutions of the heat equations under general geometric flows. Our results generalize the recent work of S. Liu, who established similar results for the Ricci flow. Both results can also be considered as the generalization of P. Li, S. T. Yau, and J. Li's gradient estimates under geometric flow setting. We also give an application to the mean curvature flow.

引用

页码：489 / 510

页数：22

共 22 条

[1] AN EXTENSION OF HOPF,E. MAXIMUM PRINCIPLE WITH AN APPLICATION TO RIEMANNIAN GEOMETRY [J].

CALABI, E .

DUKE MATHEMATICAL JOURNAL, 1958, 25 (01) :45-56

[2] Differential Harnack estimates for backward heat equations with potentials under the Ricci flow [J].

Cao, Xiaodong .

JOURNAL OF FUNCTIONAL ANALYSIS, 2008, 255 (04) :1024-1038

[3] DIFFERENTIAL HARNACK ESTIMATES FOR TIME-DEPENDENT HEAT EQUATIONS WITH POTENTIALS [J].

Cao, Xiaodong ;

Hamilton, Richard S. .

GEOMETRIC AND FUNCTIONAL ANALYSIS, 2009, 19 (04) :989-1000

[4] Mean curvature flow of surface in 4-manifolds [J].

Chen, JY ;

Li, JY .

ADVANCES IN MATHEMATICS, 2001, 163 (02) :287-309

[5] Local monotonicity and mean value formulas for evolving Riemannian manifolds [J].

Ecker, Klaus ;

Knopf, Dan ;

Ni, Lei ;

Topping, Peter .

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 2008, 616 :89-130

[6]

Guenther CM., 2002, J. Geom. Anal, V12, P425, DOI [10.1007/BF02922048, DOI 10.1007/BF02922048]

[7]

HAMILTON R.S, 1995, SURVEYS DIFFERENTIAL, VII, P7

[8]

HAMILTON RS, 1993, J DIFFER GEOM, V37, P225

[9]

HAMILTON RS, 1995, J DIFFER GEOM, V41, P215

[10]

HAMILTON RS, 1982, J DIFFER GEOM, V17, P255

← 1 2 3 →