We study the motion of smooth, closed, strictly convex hypersurfaces in
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
σ
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
φ
h
1
-
p
σ
k
=
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
φ
1
p
+
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
<
n
, we will give an example that spherical hessian of
φ
1
p
+
k
-
1
is negative definite at some point and the solution to the flow loses its smoothness.
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Christoffel-Minkowski flows Bryan, Paul; Ivaki, Mohammad; Scheuer, Julian
Transactions of the American Mathematical Society,
04/2023, Volume:
376, Issue:
4
Journal Article
Peer reviewed
Open access
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.
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.
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Abstract 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.
Orlicz–Minkowski flows Bryan, Paul; Ivaki, Mohammad N.; Scheuer, Julian
Calculus of variations and partial differential equations,
02/2021, Volume:
60, Issue:
1
Journal Article
Peer reviewed
Open access
We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if they exist, solve the regular Orlicz–Minkowski problems. ...As an application, we obtain old and new existence 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.
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Mean Convex Mean Curvature Flow with Free Boundary Edelen, Nick; Haslhofer, Robert; Ivaki, Mohammad N. ...
Communications on pure and applied mathematics,
April 2022, 2022-04-00, 20220401, Volume:
75, Issue:
4
Journal Article
Centro-affine invariants for smooth convex bodies Int. Math. Res. Notices. DOI 10.1093/imrn/rnr110, 2012 Stancu introduced a family of centro-affine normal flows, p 1\leq p>< topology.>
It is known that the Lp-curvature of a smooth, strictly convex body in Rn is constant only for origin-centered balls when 1≠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≥1, we prove a global stability result: if the Lp-curvature is almost a constant, then the volume symmetric difference of K˜ and a translate of the unit ball B is almost zero. Here K˜ is the dilation of K with the same volume as the unit ball. For 0≤p<1, we prove a similar result in the class of origin-symmetric bodies in the L2-distance. In addition, for −n<p<0, we prove a local stability result: There is a neighborhood of the unit ball that any smooth, strictly convex body in this neighborhood with ‘almost’ constant Lp-curvature is ‘almost’ the unit ball. For p=−n, we prove a global stability result in R2 and a local stability result for n>2 in the Banach-Mazur distance.
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We prove that the unit sphere is the only smooth, strictly convex solution to the isotropic Lp dual Minkowski problem hp−1|Dh|n+1−qK=1, provided (p,q)∈(−n−1,−1×n,n+1).
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