Differential Harnack estimates on manifolds--《Scientia Sinica(Mathematica)》2013年05期
Differential Harnack estimates on manifolds
NIU YanYan Department of Mathematics,University of California,San Diego
In the thesis,we study the differential Harnack estimate for the heat equation of the Hodge Laplacian deformation of(p,p)-forms on both fixed and evolving(by K?hler-Ricci flow) K?hler manifolds,which generalize the known differential Harnack estimates for(1,1)-forms.On a K?hler manifold,we define a new curvature cone C p and prove that the cone is invariant under K?hler-Ricci flow and that the cone ensures the preservation of the nonnegativity of the solutions to Hodge Laplacian heat equation.After identifying the curvature conditions,we prove the sharp differential Harnack estimates for the positive solution to the Hodge Laplacian heat equation.We also prove a nonlinear version coupled with the K?hler-Ricci flow after obtaining some interpolating matrix differential Harnack type estimates for curvature operators between Hamilton's and Cao's matrix Harnack estimates.Similarly,we define another new curvature cone C p,which is invariant under Ricci flow,and prove another interpolating matrix differential Harnack estimates for curvature operators on Riemannian manifolds.
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WANG Y;[J];Scientia Sinica(Mathematica);2014-03
【Co-citations】
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FANG Shou-wen Center of Mathematical Sciences,Zhejiang University,Hangzhou 310027,C[J];高校应用数学学报B辑(英文版);2009-03
Xu Qian FAN Department of Mathematics,Ji'nan University,Guangzhou 510632,P.R.C[J];数学学报(英文版);2007-04
Yun Yan YANG Department of Mathematics, Information School, Renmin University of China, Beijing 100872, P. R. C[J];数学学报(英文版);2010-06
Li MA;Department of Mathematics, He'nan N;[J];数学学报(英文版);2014-10
ZHAO Cheng-bing(Dept.of Mathematics,Anhui University of Architecture,Hefei 230022,China);[J];Journal of Hefei University of Technology(Natural Science);2011-04
CHEN JIE-CHENG WANG SI-LEI(Department of Mathematics, Hangzhou University, Hangzhou 310028, PRC);[J];中国科学A辑(英文版);1990-04
Jie WANG Center of Mathematical Sciences,Zhejiang University,Hangzhou 310027,C[J];中国科学(A辑:数学)(英文版);2007-11
WANG FengYu 1,2 1 School of Mathematical Sciences,Beijing Normal University,Beijing
Department of Mathematics,Swansea University,Singleton Park,Swansea SA2 8PP,UK;[J];中国科学:数学(英文版);2010-04
CHEN BingLong Department of Mathematics,Sun Yat-Sen University,Guangzhou 510275,C[J];中国科学:数学(英文版);2012-05
ZHAO Cheng-bing(Department of Math.,Anhui Institute of Architecture Industry,Hefei 230022,China);[J];Journal of Jiamusi University(Natural Science Edition);2010-03
【Co-references】
Chinese Journal Full-text Database
ZHANG QiaoFu & CUI JunZhi LSEC,ICMSEC,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,C[J];中国科学:数学(英文版);2013-08
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This is a short survey paper based on the topics of the au-
thor's research along with his collaborators. Main topics presented in
the paper are on the regularity and rigidity theorems for the solutions of
the elliptic differential equations. In particular, the author poses several
open problems in these topics for further study. The paper contains
six sections. Regularity theorems for the elliptic equations of the non-
divergence form with rough coefficients are presented in Section 1. In
Section 2, we introduce and summarize some recent developments on
the rigidity problems and theorems for the solution of some linear de-
generate elliptic equations. A typical example is the rigidity problem
for the solution of the Dirichlet problem of the Laplace-Beltrami op-
erator on the unit ball in $C^n$ with Bergman metric. In Section 3, the
rigidity theorems and problems for harmonic maps between two com-
plete non-compact Kahler manifolds are discussed. In Section 4, we
summarize the approximation formula for the potential function of the
Kahler-Einstein metric. In Section 5, we summarize some rigidity theo-
rems for the degenerate Monge-Ampère equations as well as some characterization theorems
for some strictly pseudoconvex pseudo-Hermitian
manifolds. Finally, in Section 6, we summarize some recent results on
the bottom of the spectrum of the Laplace-Beltrami operators on Kahler
manifolds.
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"... ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li–Yau’s gradient estimate. As a by-product we obtain the sharp
ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li–Yau’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy ’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li–Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n. In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman’s entropy in the case of Riemann surfaces. 1. The relation with Li–Yau’s gradient estimates In this section we provide another derivation of Theorem 1.1 of [8] and discuss its relation with Li–Yau’s gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash’s ‘entropy quantity ’
- ? M H log Hdvin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of [9]. Let u(x, t) be a positive solution to ? ? ?t -
= 0 with ?
"... is Abstract. This paper studies normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically -1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature -1. A relative estimate of Green’s function
is Abstract. This paper studies normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically -1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature -1. A relative estimate of Green’s function is proved as a tool.}