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

Abstract

Abstract. We prove that all entire smooth strictly convex self-shrinking solutions on R n to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to DingXin [5]. Moreover, we show that our argument works for a larger class of equations. In particular, we obtain rigidity results for entire selfshrinking solutions on C n to the Kähler-Ricci flow under certain conditions.

A rigid theorem for deformed Hermitian-Yang-Mills equation

“…Remark 3.2. We should also remark that the rigid theorem for a class of fully nonlinear second elliptic operator is proved in Theorem 1.2 [30]. However, the operator in [30] acts on the real symmetric matrices.…”

“…We should also remark that the rigid theorem for a class of fully nonlinear second elliptic operator is proved in Theorem 1.2 [30]. However, the operator in [30] acts on the real symmetric matrices. Using the canonical relation between Hermitian matrices and real symmetric matrices, we can regard the operator f acting on n×n Hermitian matrices as an operator f acting on 2n×2n real symmetric matrices.…”

“…This type of rigid results for self-shrinkers to parabolic flow have been studied in [2,9,10,16,29,30]. In particular, Chau-Chen-Yuan proved the rigid theorem for entire smooth Lagrangian self-shrinkers over R n .…”

“…λ i (B + J • B • J T ) complex structure on C n . In particular, the new operator f does not satisfies condition (iii) of Theorem 1.2 in[30], i.e. for any positiveB D f (B) • B ≤ k 2 , where D f (B) = ( ∂ f ∂Bij ),• is a fixed norm on the vector space of the matrices and k 2 is a certain constant.…”

On the entire self-shrinking solutions to Lagrangian mean curvature flow II

“…Later in [6], considering the drift Laplacian operator introduced by Colding-Minicozzi [5], Ding-Xin used the integral method and gave a complete improvement by dropping additional decay assumptions. In [14], the third author reproved Ding-Xin's optimal result via a pointwise approach, which also works for a larger class of equations including Hessian quotient type. If τ = π 2 , (1.4) becomes the special Lagrangian type equation…”