2,377 research outputs found
Constructing soliton solutions of geometric flows by separation of variables
This note surveys and compares results on the separation of variables
construction for soliton solutions of curvature equations including the
K\"ahler-Ricci flow and the Lagrangian mean curvature flow. In the last
section, we propose some new generalizations in the Lagrangian mean curvature
flow case.Comment: Contribution to Special Issue(s) in the Bulletin of Institute of
Mathematics, Academia Sinica (N.S.
Long-time Existence and Convergence of Graphic Mean Curvature Flow in Arbitrary Codimension
Let f:\Sigma_1 --> \Sigma_2 be a map between compact Riemannian manifolds of
constant curvature. This article considers the evolution of the graph of f in
the product of \Sigma_1 and \Sigma_2 by the mean curvature flow. Under suitable
conditions on the curvature of \Sigma_1 and \Sigma_2 and the differential of
the initial map, we show that the flow exists smoothly for all time. At each
instant t, the flow remains the graph of a map f_t and f_t converges to a
constant map as t approaches infinity. This also provides a regularity estimate
for Lipschtz initial data.Comment: to be published in Inventiones Mathematica
Quasilocal mass and surface Hamiltonian in spacetime
We discuss the concepts of energy and mass in relativity. On a finitely
extended spatial region, they lead to the notion of quasilocal energy/mass for
the boundary 2-surface in spacetime. A new definition was found in [27] that
satisfies the positivity, rigidity, and asymptotics properties. The definition
makes use of the surface Hamiltonian term which arises from Hamilton-Jacobi
analysis of the gravitation action. The reference surface Hamiltonian is
associated with an isometric embedding of the 2-surface into the Minkowski
space. We discuss this new definition of mass as well as the reference sur-
face Hamiltonian. Most of the discussion is based on joint work with PoNing
Chen and Shing-Tung Yau.Comment: 11 pages, contribution to ICMP 201
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