Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under th...Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.展开更多
Monotonicity formulas play a central role in the study of free boundary problems.In this note,we develop a Weiss-type monotonicity formula for solutions to parabolic free boundary problems on metric measure cones.
基金supported by NSFC (10831008)NKBRPC(2006CB805905)
文摘Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.
基金The second author was partially supported by National Key R&D Program of China(2021YFA1003001)NSFC 12025109,and the third author was partially supported by NSFC(11521101).
文摘Monotonicity formulas play a central role in the study of free boundary problems.In this note,we develop a Weiss-type monotonicity formula for solutions to parabolic free boundary problems on metric measure cones.
