We establish a special concavity property for positive Hessian quotient operators sigma(n)(W)/sigma(n-k)(W), 1 <= k <= n-1. As a consequence, we prove a Jacobi inequality for a general symmetric tensor satisfying a positive Hessian quotient equation on Riemannian manifolds.

We prove that the normalized second fundamental form A of immersed C^2 hypersurface in a space form (N^n+1, g̅) is intrinsic provided σ _2k+1(A) 0 for some k≥ 1 . We also establish the intrinsicality of the normalized second fundamental form A of M^n=∂Ω for domain Ω⊂ N^n+1, n≥ 3 .

We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in Rn+1 ${\mathbb{R}}^{n+1}$ , and prove the existence and regularity of the flow before extincting to a point in finite time.

We extend the weighted gradient estimate for solutions of nonlinear PDE associated to the prescribed $ k $-th $ L^p $-area measure problem to the case $ 0 < p < 1 $. The estimate yields non-collapsing estimate for symmetric convex bodied with prescribed $ L^p $-area measures.

We provide a natural simple argument using anistropic flows to prove the existence of weak solutions to Lutwak's $L^p$-Minkowski problem on $S^n$ which were obtained by other methods.

We establish a necessary and sufficient condition for C1," regularity of the admissible square root of a non-negative C-2,C-2a(R) function.

This expository paper presents the current knowledge of particular fully nonlinear curvature flows with local forcing term, so-called locally constrained curvature flows. We focus on the spherical ambient space. The flows are designed to preserve a quermassintegral and to de-/increase the other quermassintegrals. The convergence of this flow to a round sphere would settle the full set of quermassintegral inequalities for convex domains of the sphere, but a full proof is still missing. Here we collect what is known and hope to attract wide attention to this interesting problem.

New types of hypersurface flows have been introduced recently with goals to establish isoperimetric type inequalities in geometry.These flows serve as efficient paths to achieve the optimal solutions to the problems of calculus of variations in geometric setting.The main idea is to use variational structures to develop hypersurface flows which are monotonic for the corresponding curvature integrals (including volume and surface area).These new geometric flows pose interesting but challenging PDE problems.Resolution of these problems have significant geometric implications.

In this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel-Minkowski problem in convex geometry. We will also study the Dirichlet problem of the corresponding degenerate equations as an extension of the equations studied by Krylov.

In this article, we continue the work in [Int. Math. Res. Not. IMRN 13 (2015), pp. 4716–4740] and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.

We consider a fully nonlinear partial differential equation associated to the intermediate L^p Christoffel–Minkowski problem in the case 1<p<k+1 . We establish the existence of convex body with prescribed k -th even p -area measure on 𝕊^n , under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain C^1,1 regularity estimates for admissible solutions of the equation when p≥k+1/2 .

We consider a fully nonlinear partial differential equation associated to the intermediate \(L^p\) Christoffel–Minkowski problem in the case \(1<p<k+1\). We establish the existence of convex body with prescribed k-th even p-area measure on \(\mathbb S^n\), under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain \(C^{1,1}\) regularity estimates for admissible solutions of the equation when \( p\ge \frac{k+1}{2}\).

We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold \((N^{n+1}, \bar{g})\) through regularity study of a degenerate fully nonlinear curvature equation in general Riemannian manifold. The estimate has a direct consequence for the Weyl isometric embedding problem of \(({\mathbb {S}}^2, g)\) in 3-dimensional warped product space \((N^3, \bar{g})\). We also discuss isometric embedding problem in spaces with horizon in general relativity, like the Anti-de Sitter–Schwarzschild manifolds and the Reissner–Nordström manifolds.

In this paper we prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in $C^\infty$-topology to a smooth strictly convex soliton as $t$ approaches to infinity is obtained as a consequence of these estimates together with an earlier result of Andrews. The estimates are established via the study of a new entropy functional for the flow.

The electronic and optical properties of strained monolayer arsenene were calculated based on first-principle density functional theory. Our theoretical calculations demonstrated that monolayer arsenene was transformed from indirect to direct band gap semiconductor by inducing uniaxial tensile strain along armchair and zigzag directions. Compared to the biaxial tensile strain of 0.04, this transformation occurred at the strain of 0.06 and 0.10 along armchair and zigzag direction, respectively. Spin-orbital coupling is available to tune the bandgaps. The spin-orbit interaction opens a 0.2 eV bandgap in the Gamma-point on the unstrained monolayer arsenene. The absorption properties were calculated and a clear red shift was observed with the increasing strain.

We give a new proof of a classical uniqueness theorem of Alexandrov [4] using the weak uniqueness continuation theorem of Bers–Nirenberg [8]. We prove a version of this theorem with the minimal regularity assumption: the spherical Hessians of the corresponding convex bodies as Radon measures are nonsingular.

We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α>1n+2 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere for α≥1.

Heterogeneity is commonly believed to be intrinsic to metallic glasses (MGs). Nevertheless, how to distinguish and characterize the heterogeneity at the atomic level is still debated. Based on the extensive molecular dynamics simulations that combine isoconfigurational ensemble and atomic pinning methods, we directly reveal that MG contains flow units and the elastic matrix which can be well distinguished by their distinctive atomic-level responsiveness and mechanical performance. The microscopic features of the flow units, such as the shape, spatial distribution dimensionality, and correlation length, are characterized from atomic position analyses. Furthermore, the correlation between the flow units and the landscape of energy state, free volume, atomic-level stress, and especially the local bond orientational order parameter is discussed.