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We prove that if the density of the cone-volume measure of a smooth, strictly convex body with respect to the spherical Lebesgue measure is nearly constant, a homothetic copy of the body is close to the unit ball in the $L^2$-distance.

In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space $\mathbb{R}^{n+1}$, \begin{align*} \dot{x}=\left(\frac{1}{\frac{E_k(\hat{\kappa})}{E_{k-1}(\hat{\kappa})}-\alpha }-\langle x,\nu\rangle\right)\nu, \quad k=2,3,\ldots,n-1. \end{align*} Assuming that the initial hypersurface $\mathcal{M}_0 \subset \mathbb{R}^{n+1}$ is star-shaped and its shifted principal curvatures $\hat{\kappa}=\kappa+\alpha(1,\ldots,1)$ lie in the convex set \begin{align*} \Gamma_{\alpha,k}:=\Gamma_{k-1}\cap \{\lambda\in \mathbb{R}^n:\, E_k(\lambda)-\alpha E_{k-1}(\lambda)>0\}, \end{align*} we show that the flow admits a smooth solution that exists for all positive times, and it converges smoothly to a round sphere. As a corollary, we obtain a new set of Alexandrov-Fenchel-type inequalities for non-convex domains. In the second part, we derive a Poincar\'{e} type inequality for $k$-convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality.

In this paper, we establish the existence of smooth, origin-symmetric, strictly convex solutions to the prescribed even $L_p$ curvature problem., Comment: to appear in Advances in Mathematics

We prove that the unit sphere is the only smooth, strictly convex solution to the isotropic $L_p$ dual Minkowski problem \begin{align*} h^{p-1} |D h|^{n+1-q}\mathcal{K}=1, \end{align*} provided $(p,q)\in (-n-1,-1]\times [n,n+1)$., Comment: 10 pages

Let $K$ be a smooth, origin-symmetric, strictly convex body in $\mathbb{R}^n$. If for some $\ell\in GL(n,\mathbb{R})$, the anisotropic Riemannian metric $\frac{1}{2}D^2 \Vert\cdot\Vert_{\ell K}^2$, encapsulating the curvature of $\ell K$, is comparable to the standard Euclidean metric of $\mathbb{R}^{n}$ up-to a factor of $\gamma > 1$, we show that $K$ satisfies the even $L^p$-Minkowski inequality and uniqueness in the even $L^p$-Minkowski problem for all $p \geq p_\gamma := 1 - \frac{n+1}{\gamma}$. This result is sharp as $\gamma \searrow 1$ (characterizing centered ellipsoids in the limit) and improves upon the classical Minkowski inequality for all $\gamma < \infty$. In particular, whenever $\gamma \leq n+1$, the even log-Minkowski inequality and uniqueness in the even log-Minkowski problem hold., Comment: 19 pages. Final version, to appear in International Mathematics Research Notices

We show that the only even, smooth, convex solutions to a class of isotropic mixed Christoffel-Minkowski type problems are origin-centred spheres, which, in particular, answers a question of Firey 74 in the even isotropic case about kinematic measures. Employing the Heintze-Karcher inequality, we prove that the only smooth, strictly convex solutions to a large class of Minkowski type problems are origin-centred spheres. Immediate corollaries are the uniqueness of solutions to the isotropic Orlicz-Minkowski problem and the isotropic $L_p$-Gaussian-Minkowski problem when $p\geq 1$.

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 non-linearities 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., Comment: 11 pages

It is known that the $L_p$-curvature of a smooth, strictly convex body in $\mathbb{R}^{n}$ is constant only for origin-centred balls when $1\neq p>-n$, and only for balls when $p=1$. If $p=-n$, then the $L_{-n}$-curvature is constant only for origin-symmetric ellipsoids. We prove `local' and `global' stability versions of these results. For $p\geq 1$, we prove a global stability result: if the $L_p$-curvature is almost a constant, then the volume symmetric difference of $\tilde{K}$ and a translate of the unit ball $B$ is almost zero. Here $\tilde{K}$ is the dilation of $K$ with the same volume as the unit ball. For $0\leq p<1$, we prove a similar result in the class of origin-symmetric bodies in the $L^2$-distance. In addition, for $-n2$ in the Banach-Mazur distance.

An important set of theorems in geometric analysis consists of constant rank theorems for a wide variety of curvature problems. In this paper, for geometric curvature problems in compact and non-compact settings, we provide new proofs which are both elementary and short. Moreover, we employ our method to obtain constant rank theorems for homogeneous and non-homogeneous curvature equations in new geometric settings. One of the essential ingredients for our method is a generalization of a differential inequality in a viscosity sense satisfied by the smallest eigenvalue of a linear map (Brendle-Choi-Daskalopoulos, Acta Math. 219(2017): 1-16) to the one for the subtrace. The viscosity approach provides a concise way to work around the well known technical hurdle that eigenvalues are only Lipschitz in general. This paves the way for a simple induction argument., Comment: 22 pages

We provide a curvature flow approach to the regular Christoffel-Minkowski problem. The speed of our curvature flow is of an entropy preserving type and contains a global term., Comment: 25 pages

We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if exist, solve the regular Orlicz-Minkowski problems. As an application, we obtain old and new results for the regular even Orlicz-Minkowski problems; the corresponding $L_p$ version is the even $L_p$-Minkowski problem for $p>-n-1$. Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz-Minkowski problems; the $L_p$ versions are the even $L_p$-Minkowski problem for $p>0$ and the $L_p$-Minkowski problem for $p>1$. In the final section, we use a curvature flow with no global term to solve a class of $L_p$-Christoffel-Minkowski type problems., Comment: 30 pages

We employ curvature flows without global terms to seek strictly convex, spacelike solutions of a broad class of elliptic prescribed curvature equations in the simply connected Riemannian spaceforms and the Lorentzian de Sitter space, where the prescribed function may depend on the position and the normal vector. In particular, in the Euclidean space we solve a class of prescribed curvature measure problems, intermediate $L_p$-Aleksandrov and dual Minkowski problems as well as their counterparts, namely the $L_{p}$-Christoffel-Minkowski type problems. In some cases we do not impose any condition on the anisotropy except positivity, and in the remaining cases our condition resembles the constant rank theorem/convexity principle due to Caffarelli-Guan-Ma (Commun. Pure Appl. Math. 60 (2007), 1769--1791). Our approach does not rely on monotone entropy functionals and it is suitable to treat curvature problems that do not possess variational structures., Comment: 32 pages

We study the iterations of a class of curvature image operators $\Lambda_p^{\varphi}$ introduced by the author in (J. Funct. Anal. 271 (2016) 2133--2165). The fixed points of these operators are the solutions of the $L_p$ Minkowski problems with the positive continuous prescribed data $\varphi$. One of our results states that if $p\in (-n,1)$ and $\varphi$ is even, or if $p\in (-n,-n+1]$, then the iterations of these operators applied to suitable convex bodies sequentially converge in the Hausdorff distance to fixed points., Comment: minor revisions

In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish such a bound via the maximum principle for a triple-approximation scheme, which combines ideas from Edelen, Haslhofer-Hershkovits, and Volkmann. Other important new ingredients are a Bernstein-type theorem and a sheeting theorem for low entropy free boundary flows in a halfslab, which allow us to rule out multiplicity 2 (half-)planes as possible tangent flows and, for mean convex domains, as possible limit flows., Comment: 52 pages

This short note is a mostly expository article examining negatively curved three-manifolds. We look at some rigidity properties related to isometric embeddings into Minkowski space. We also review the Cross Curvature Flow (XCF) as a tool to study the space of negatively curved metrics on hyperbolic three-manifolds, the largest and least understood class of model geometries in Thurston's Geometrisation. The relationship between integrability and embedability yields interesting insights, and we show that solutions with fixed Einstein volume are precisely the integrable solutions, answering a question posed by Chow and Hamilton when they introduced the XCF.

We study the motion of smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ expanding in the direction of their normal vector field with speed depending on the $k$th elementary symmetric polynomial of the principal radii of curvature $\sigma_k$ and support function $h$. A homothetic self-similar solution to the flow that we will consider in this paper, if exists, is a solution of the well-known $L_p$-Christoffel-Minkowski problem $\varphi h^{1-p}\sigma_k=c$. Here $\varphi$ is a preassigned positive smooth function defined on the unit sphere, and $c$ is a positive constant. For $1\leq k\leq n-1, p\geq k+1$, assuming the spherical hessian of $\varphi^{\frac{1}{p+k-1}}$ is positive definite, we prove the $C^{\infty}$ convergence of the normalized flow to a homothetic self-similar solution. One of the highlights of our arguments is that we do not need the constant rank theorem/deformation lemma of Guan-Ma and thus we give a partial answer to a question raised in Guan-Xia. Moreover, for $k=n, p\geq n+1$, we prove the $C^{\infty}$ convergence of the normalized flow to a homothetic self-similar solution without imposing any further condition on $\varphi.$ In the final section of the paper, for $1\leq k

In this short note, we prove that on the three-sphere with any bumpy metric there exist at least four solutions of the Allen-Cahn equation with spherical interface and index at most two. The proof combines several recent results from the literature.

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifold of non-negative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequalities, so-called "pseudo"-Harnack inequalities, for expanding flows in the sphere and in the hyperbolic space.

We provide partial answers to the open problems 4.5, 4.6 of \cite{Gardner} and 12.9 of \cite{Lut1} regarding the classification of fixed points of the second mixed projection operator and iterates of the projection and centroid operators., Comment: minor changes,to appear in JFA

Employing the inverse function theorem on Banach spaces, we prove that in a $C^{2}(S^{n-1})$-neighborhood of the unit ball, the only solutions of $\Pi^2K=cK$ are origin-centered ellipsoids. Here $K$ is an $n$-dimensional convex body, $\Pi K$ is the projection body of $K$ and $\Pi^2K=\Pi(\Pi K).$, Comment: to appear in Mathematika

We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even $L_p$-Minkowski problems in $\mathbb{R}^{n+1}$ for $p>-n-1.$, Comment: 21 pages. Comments are welcome

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the notion of `quasi-ancient' solutions for flows that do not admit non-trivial, convex, ancient solutions. Such solutions are somewhat analogous to ancient solutions for flows such as the mean curvature flow, or 1-homogeneous flows. The techniques presented here allow us to prove that any convex, quasi-ancient solution of a curvature flow which satisfies a backwards in time uniform bound on mean curvature must be stationary or a family of shrinking geodesic spheres. The main tools are geometric, employing the maximum principle, a rigidity result in the sphere and an Alexandrov reflection argument. We emphasize that no homogeneity or convexity/concavity restrictions are placed on the speed, though we do also offer a short classification proof for several such restricted cases., Comment: 20 pages, 1 figure. Slight improvements added to section 6. Comments are welcome

We prove the asymptotic roundness under normalized Gauss curvature flow provided entropy is initially small enough.

We prove Harnack inequalities for hypersurfaces flowing on the unit sphere by $p$-powers of a strictly monotone, 1-homogeneous, convex, curvature function $f$, $0

We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by their mean curvature. We prove a differential Harnack inequality for any weakly convex solution to the mean curvature flow. As an application, by applying an Aleksandrov reflection argument, we classify convex, ancient solutions of the mean curvature flow on the sphere., Comment: to appear in Asian J. Math

We study the motion of smooth, strictly convex bodies in $\mathbb{R}^n$ expanding in the direction of their normal vector field with speed depending on Gauss curvature and support function., Comment: (to appear) J. Funct. Anal

We prove that the only compact convex ancient solutions of the planar affine normal flow are contracting ellipses., Comment: J. Geom. Anal. (to appear)

Using polar convex bodies and the $C_0$-bounds from Guan and Ni \cite{PL}, we obtain a uniform lower bound on the Gauss curvature of the normalized solution of the Gauss curvature flow without using Chow's Harnack inequality \cite{Ch2}., Comment: to appear in Indiana Univ. Math. J

We give a stability version of of the Blaschke-Santal\'{o} inequality in the plane., Comment: Final version: updated references. To appear in Monatshefte f\"ur Mathematik

In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of smooth, origin-symmetric convex bodies. First, we prove that the $\infty$-flow evolves suitably normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo $SL(2).$ Second, using the $\infty$-flow and a Harnack estimate for this flow, we prove a stability version of the planar Busemann-Petty centroid inequality in the Banach-Mazur distance. Third, we prove that the convergence of normalized solutions in the Hausdorff metric can be improved to convergence in the $\mathcal{C}^{\infty}$ topology.

By means of dual convex bodies, we obtain regularity of solutions to the expanding Gauss curvature flows with homogeneity degrees $-p$, $0

We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke-Santal\'{o} inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitably rescaled solutions converge sequentially to the unit ball in the $\mathcal{C}^{\infty}$ topology modulo $SL(n+1)$., Comment: Calc. Var. PDE. (to appear)

We study the long time behavior of the volume preserving $p$-flow in $\mathbb{R}^{n+1}$ for $1\leq p<\frac{n+1}{n-1}$. By extending Andrews' technique for the flow along the affine normal, we prove that every centrally symmetric solution to the volume preserving $p$-flow converges sequentially to the unit ball in the $C^{\infty}$ topology, modulo the group of special linear transformations., Comment: Fixed a sign error

We prove that the only compact, origin-symmetric, strictly convex ancient solutions of the planar $p$ centro-affine normal flows are contracting origin-centered ellipses., Comment: I changed the title and fixed some typos. To appear in Annales de l'Institut Henri Poincar\'e (C) Analyse Non Lin\'eaire

Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric convex body in $\mathbb{R}^2$ such that it has area $\pi$ and its $p$-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, $K$ is close to an ellipse in the Hausdorff distance., Comment: Fixed typos, to appear in J. Geom. Anal

A generalization of the affine-geometric Wirtinger inequality for curves to hypersurfaces is given.

We consider two types of $p$-centro affine flows on smooth, centrally symmetric, closed convex planar curves, $p$-contracting, respectively, $p$-expanding. Here $p$ is an arbitrary real number greater than 1. We show that, under any $p$-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area $\pi$ converge, in the Hausdorff metric, to the unit circle modulo SL(2). As a $p$-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any $p$-expanding flow, curves expand to infinity in finite time, while the only homothetic solutions of the flow are ellipses centered at the origin. If the curves are normalized as to enclose constant area $\pi$, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to SL(2) transformations, to the unit circle. At the end, we present a new proof of $p$-affine isoperimetric inequality, $p\geq 1$, for smooth, centrally symmetric convex bodies in $\mathbb{R}^2$., Comment: to appear in Trans. Amer. Math. Soc. arXiv admin note: text overlap with arXiv:1205.6455

We prove that the set of smooth, $\pi$-periodic, positive functions on the unit circle for which the $L_{-2}$ Minkowski problem is solvable is dense in the set of all smooth, $\pi$-periodic, positive functions on the unit circle with respect to the $L^{\infty}$ norm. Furthermore, we obtain a necessary condition on the solvability of the even $ L_{-2}$ Minkowski problem. At the end, we prove uniqueness of the solutions up to an affine linear transformation., Comment: to appear in Adv. Appl. Math, fixed some typos