Uniqueness of closed self-similar solutions to $σ_k^{α}$-curvature flow

Employing a local version of the Brunn-Minkowski inequality, we give a new and simple proof of a result due to Andrews, Choi and Daskalopoulos that the origin-centred balls are the only closed, self-similar solutions of the Gauss curvature flow. Extensions to various nonlinearities are obtained, assuming the centroid of the enclosed convex body is at the origin. By applying our method to the Alexandrov-Fenchel inequality, we also show that origin-centred balls are the only solutions to a large class of even Christoffel-Minkowski type problems. 

Uniqueness of solutions to a class of isotropic curvature problems

A recent article (Li and Lv, J Geom Anal 30:417–447, 2020) considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we consider self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting self-similarly are necessarily spheres.

Contracting Self-similar Solutions of Nonhomogeneous Curvature Flows

In this paper, we consider a family of closed hypersurfaces which shrink self-similarly with speed of quotient curvatures. We show that the only such hypersurfaces are shrinking spheres.

Uniqueness of self-similar solutions to flows by quotient curvatures

Abstract In this paper, we investigate closed strictly convex hypersurfaces in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {\mathbb{R}^{n+1}} which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree 1 and inverse concave function of the principal curvatures with power greater than 1, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>𝕊</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> {\mathbb{S}^{n+1}_{+}} with power greater than or equal to 1. 

Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p < 1

$L_p$-Christoffel-Minkowski problem arises naturally in the $L_p$-Brunn-Minkowski theory. It connects both curvature measures and area measures of convex bodies and is a fundamental problem in convex geometric analysis. Since the lack of Firey's extension of Brunn-Minkowski inequality and constant rank theorem for $p<1$, the existence and uniqueness of $L_p$-Brunn-Minkowski problem are difficult problems. In this paper, we prove a uniqueness theorem for solutions to $L_p$-Christoffel-Minkowski problem with $p<1$ and constant prescribed data. Our proof is motivated by the idea of Brendle-Choi-Daskaspoulos's work on asymptotic behavior of flows by powers of the Gaussian curvature. One of the highlights of our arguments is that we introduce a new auxiliary function $Z$ which is the key to our proof.

Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p<1

In this paper, we study a class of doubly nonlinear parabolic equations (1.1). The nonlinearity β(u)$$ \beta (u) $$ brings great difficulties to the study of the problem. First, we show that the problem has a unique solution. Then, we prove that the process corresponding to the problem is norm‐to‐weak continuous. After that, by using Legendre transform, we obtain uniform estimates and asymptotic compactness properties that allow us to ensure the existence of pullback D$$ \mathcal{D} $$ ‐attractors for the associated process to the problem.

Pullback 
 D$$ \mathcal{D} $$ ‐attractors for doubly nonlinear parabolic equations

Recently, the horospherical $p$-Minkowski problem in hyperbolic space was proposed as a counterpart of $L_p$ Minkowski problem in Euclidean space. Through designing a new volume preserving curvature flow, the existence of normalized even solution to the horospherical $p$-Minkowski problem was solved for all $p \in \mathbb{R}$. However, due to the lack of homogeneity of the horospherical $p$-surface area measure, it is difficult to remove the normalizing factor. In this paper, we overcome this difficulty for $-n\leq p

Non-normalized solutions to the horospherical Minkowski problem

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Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p < 1

Uniqueness of solutions to Lp-Christoffel-Minkowski problem for p<1

Pullback D$$ \mathcal{D} $$ ‐attractors for doubly nonlinear parabolic equations

Non-normalized solutions to the horospherical Minkowski problem