Rigidity of entire self-shrinking solutions to curvature flows

Abstract

We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in C m with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in C m with the pseudo-Euclidean metric is flat if the Hessian of the potential is bounded below quadratically; and (c) the Hermitian counterpart of (b) for the Kähler Ricci flow.

Rigidity of Entire Convex Self-Shrinking Solutions to Hessian Quotient Flows

“…The common part of the arguments in [3], [5] and here is proving the constancy of a natural quantity, the phase φ = ln det D 2 u (φ = ln q n 1 ,n 2 (D 2 u) in the Hessian quotient case). Then the homogeneity of the self-similar term on the right-hand side of the equation leads to the quadratic conclusion.…”

“…[4,8,10,11]). Rigidity of entire smooth convex solutions to (1.3) has been studied in [3,5,8,9]. In [3] and [9], the authors proved that any smooth convex solution to (1.3) must be quadratic under the condition that the Hessian is bounded below inversely quadratically.…”

“…The phase satisfies an elliptic equation without zeroth order term (shown below in (2.21)). In [3], using the inversely quadratic decay assumption, Chau-Chen-Yuan constructed a specific barrier function to force the supremum of the phase in R n to be attained at some point. Then the strong maximum principle implies the constancy of the phase.…”

Rigidity of complete spacelike translating solitons in pseudo-Euclidean space

“…For non-parametric Lagrangian solitons of mean curvature flow, there are several interesting rigidity theorems. As to entire self-shrinker for mean curvature flow in R 2n n with the indefinite metric n i=1 dx i dy i , Huang-Wang [16] and Chau-Chen-Yuan [5] used different methods to prove the rigidity of entire self-shrinker under a decay condition on the induced metric (D 2 f ). Later, using an integral method, Ding and Xin [13] removed the additional decay condition and proved any entire smooth convex self-shrinking solution for mean curvature flow in R 2n n is a quadratic polynomial.…”

On the rigidity theorems for Lagrangian translating solitons in pseudo-Euclidean space III

“…As to self similar solutions for mean curvature flow in R 2n n with the indefinite metric n i=1 dx i dx j , Huang-Wang [8] and Chau-Chen-Yuan [4] used different methods to prove the rigidity of entire self-shrinking solutions under a decay condition on the Hessian (D 2 f ). Later, using an integral method, Ding and Xin [6] removed the additional decay condition and proved any entire smooth convex self-shrinking solution for mean curvature flow in R 2n n is a quadratic polynomial.…”