Rigidity of entire self-shrinking solutions to curvature flows
Self-shrinking solutions of the mean curvature flow arise as ancient families of immersions from a Riemannian manifold \(\Sigma\) into \(\mathbb R^N\), \(F(x,t): \Sigma \times (-\infty, 0) \rightarrow \mathbb R^{N}\), that evolve only by scaling. In that case \(F(x,-1)\) satisfies the equation NEWLINE\[NEWLINE H+\frac{1}{2} F^\perp = 0, NEWLINE\]NEWLINE where \(H\) is the mean curvature of \(F(\Sigma,t)\) and \(\perp\) is the normal component. A Lagrangian graph solution is one of the form \(\{ (x,Du(x)): x \in \mathbb R^n \}\). The authors prove that any entire graphic self-shrinking solution of the Lagrangian mean curvature flow in \(\mathbb C^{n}\) with the Euclidean metric is flat. They then equip \(\mathbb R^{2n}\) with the indefinite metric \(\sum_{i=1}^n dx_idy_i,\) in which case the potential \(u\) of a space-like graphic self-shrinking solution is convex and satisfies an elliptic equation. They prove that if \(u\) is either radially symmetric or if its Hessian is bounded below quadratically, then the solution is flat. They further prove a Hermitian analog of this result for the Kähler Ricci flow.
scientific article; zbMATH DE number 6022599