完备非紧黎曼流形上的极值原理
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摘要
本文将热方程的次解估计推广至具有低阶项的热型方程的次解估计,并讨论了张量型的极值原理及向量丛上的Weinberger-Hamilton型极值原理.在Ricci流的作用下,一些曲率的发展满足热型方程或方程组,利用极值原理,可以得到这些曲率在Ricci流作用下随时间的变化情况.
In this paper we discuss the maximum principles on noncompact manifolds analogues to compact case.It is important to control the growth of the curvature as the distance tends to infinity.We extend the maximum principle to the quasi-linear versions of the heat equation.Also we discuss the tensor version of the maximum principles and Weinberger-Hamilton's maximum principles.Since the curvatures evolve as heat-type parabolic equations under Ricci flow,we discuss some applications of the maximum principles in Ricci flow at last.
In this paper we discuss the maximum principles on noncompact manifolds analogues to compact case.It is important to control the growth of the curvature as the distance tends to infinity.We extend the maximum principle to the quasi-linear versions of the heat equation.Also we discuss the tensor version of the maximum principles and Weinberger-Hamilton's maximum principles.Since the curvatures evolve as heat-type parabolic equations under Ricci flow,we discuss some applications of the maximum principles in Ricci flow at last.
引文
[1]周蜀林,偏微分方程.北京大学出版社.
[2]丘成桐,孙理察,微分几何讲义.高等教育出版社.
[3]Evans,L,Partial Differential Equations.AMS.Providence,1998.
[4]Hamilton,R.S.,Three-manifolds with positive Ricci curvature.J.Diffcrential Geom.17(1982),no.2,255-306.
[5]Hamilton,R.S.,Four-manifolds with positive curvature operators.J.Differential Geom.24(1986),no.2,153-179.
[6]Chow,B.;Knopf,D.,The Ricci flow:An introduction.Mathematical surveys and monographs,vol.110.
[7]Chow,B.;Chu,Sun_Chin;Glickenstein,D.;Guenther,C.;Isenberg,J.;Ivey,T.;Knopf,D.;Lu,Peng;Luo,Feng;Ni,lei,The Ricci flow:Techniques and applications.Mathematical surveys and monographs,vol.144.
[8]Chow,B.;Lu,Peng;Ni,Lei,Hamilton's Ricci flow.AMS.Providence,Rhode Island,2006.
[9]Shi,Wan_Xiong,Deforming the metric on complete Riemannian manifolds.J.Differential Geom.30(1989),223-301.
[10]Karp,K.;Li,P.,The heat equation on complete Riemannian manifolds,unpublished,1982.
[11]Ni,Lei;Tam,Luen_Fai,K(a|¨)hler_Ricci flow and the Poincar(?)_Lelong equation.Comm.Anal.Geom.12(2004),111-141.
[12]Ni,Lei;Wu,Baoqiang,Complete manifolds with nonnegative curvature operator.Proc.Amer.Math.Soc.135(2007),3021-3028.
[13]Ni,Lei,Ricci flow and nonnegativity of curvature.arXiv:math\0305246vl.
[14]Ni,Lei;Tam,Luen_Fai,Plurisubharmonic functions and the structure of complete K(a|¨)hler manifolds with nonnegative curvature.arXiv:math\0304096v1.
[15]Bohm,C.;Wilking,B.,Manifolds with positive curvature operators are space forms.arXiv:math\0606187v1.
[16]B(o|¨)hm,C.;Wilking,B.,Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature.GAFA.Geom,funct.anal.Vbl.l7(2007)665-681.
[2]丘成桐,孙理察,微分几何讲义.高等教育出版社.
[3]Evans,L,Partial Differential Equations.AMS.Providence,1998.
[4]Hamilton,R.S.,Three-manifolds with positive Ricci curvature.J.Diffcrential Geom.17(1982),no.2,255-306.
[5]Hamilton,R.S.,Four-manifolds with positive curvature operators.J.Differential Geom.24(1986),no.2,153-179.
[6]Chow,B.;Knopf,D.,The Ricci flow:An introduction.Mathematical surveys and monographs,vol.110.
[7]Chow,B.;Chu,Sun_Chin;Glickenstein,D.;Guenther,C.;Isenberg,J.;Ivey,T.;Knopf,D.;Lu,Peng;Luo,Feng;Ni,lei,The Ricci flow:Techniques and applications.Mathematical surveys and monographs,vol.144.
[8]Chow,B.;Lu,Peng;Ni,Lei,Hamilton's Ricci flow.AMS.Providence,Rhode Island,2006.
[9]Shi,Wan_Xiong,Deforming the metric on complete Riemannian manifolds.J.Differential Geom.30(1989),223-301.
[10]Karp,K.;Li,P.,The heat equation on complete Riemannian manifolds,unpublished,1982.
[11]Ni,Lei;Tam,Luen_Fai,K(a|¨)hler_Ricci flow and the Poincar(?)_Lelong equation.Comm.Anal.Geom.12(2004),111-141.
[12]Ni,Lei;Wu,Baoqiang,Complete manifolds with nonnegative curvature operator.Proc.Amer.Math.Soc.135(2007),3021-3028.
[13]Ni,Lei,Ricci flow and nonnegativity of curvature.arXiv:math\0305246vl.
[14]Ni,Lei;Tam,Luen_Fai,Plurisubharmonic functions and the structure of complete K(a|¨)hler manifolds with nonnegative curvature.arXiv:math\0304096v1.
[15]Bohm,C.;Wilking,B.,Manifolds with positive curvature operators are space forms.arXiv:math\0606187v1.
[16]B(o|¨)hm,C.;Wilking,B.,Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature.GAFA.Geom,funct.anal.Vbl.l7(2007)665-681.