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Let |$K$| be a smooth, origin-symmetric, strictly convex body in |${\mathbb{R}}^{n}$|⁠. If for some |$\ell \in \textrm{GL}(n,{\mathbb{R}})$|⁠ , the anisotropic Riemannian metric |$\frac{1}{2}D^{2} \left \Vert \cdot \right \Vert_{\ell K}^{2}$|⁠ , 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. [ABSTRACT FROM AUTHOR]

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 et al. (Acta Math 219:1–16, 2017) 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we generalize White's regularity and structure theory for mean‐convex mean curvature flow [45, 46, 48] 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 [9], Haslhofer‐Hershkovits [16], and Volkmann [43]. Other important new ingredients are a Bernstein‐type theorem and a sheeting theorem for low‐entropy free boundary flows in a half‐slab, which allow us to rule out multiplicity 2 (half‐)planes as possible tangent flows and, for mean‐convex domains, as possible limit flows. © 2021 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant nonnegative 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 manifolds of nonnegative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequality, so-called "pseudo"-Harnack inequality, for expanding flows in the sphere and in the hyperbolic space. [ABSTRACT FROM AUTHOR]

We study the iterations of a class of curvature image operators Λpφ introduced by Ivaki in [J. Funct. Anal. 271 (2016) 2133–2165]. The fixed points of these operators are the solutions of the Lp Minkowski problems with the positive continuous prescribed data φ. One of our results states that if p∈(−n,1) and φ is even, or if p∈(−n,−n+1], then the iterations of these operators applied to suitable convex bodies sequentially converge in the Hausdorff distance to fixed points. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of smooth, origin-symmetric, strictly convex solutions to the prescribed even L p curvature problem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

By means of dual convex bodies, we obtain regularity of solutions of the expanding Gauss curvature flows with the homogeneity degrees -p, 0 < p < 1. At the end, we remark that our method can also be used to obtain regularity of solutions to the shrinking Gauss curvature flows with the homogeneity degrees less than one. [ABSTRACT FROM AUTHOR]

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 et al. (2007). Our approach does not rely on monotone entropy functionals and it is suitable to treat curvature problems that do not possess variational structures. [ABSTRACT FROM AUTHOR]

We study the motion of smooth, closed, strictly convex hypersurfaces in Rn+1 expanding in the direction of their normal vector field with speed depending on the kth elementary symmetric polynomial of the principal radii of curvature σk 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 Lp-Christoffel-Minkowski problem φh1-pσk=c. Here φ is a preassigned positive smooth function defined on the unit sphere, and c is a positive constant. For 1≤k≤n-1,p≥k+1, assuming the spherical hessian of φ1p+k-1 is positive definite, we prove the C∞ 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 and Ma (Invent Math 151:553-577, 2003) and thus we give a partial answer to a question raised in Guan and Xia (Calc Var Part Differ Equ 57:69, 2018. https://doi.org/10.1007/s00526-018-1341-y. Moreover, for k=n,p≥n+1, we prove the C∞ convergence of the normalized flow to a homothetic self-similar solution without imposing any condition on φ. In the final section of the paper, for 1≤k