اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدVolume preserving flow and Alexandrov-Fenchel type inequalities in hyperbolic space
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In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of $k$-th mean curvatures with $k=1,cdots,n$, and positive powers of $p$-th power sums $S_p$ with $p>0$. We prove that if the initial hypersurface $M_0$ is smooth and closed and has positive sectional curvatures, then the solution $M_t$ of the flow has positive sectional curvature for any time $t>0$, exists for all time and converges to a geodesic sphere exponentially in the smooth topology. The convergence result can be used to show that certain Alexandrov-Fenchel quermassintegral inequalities, known previously for horospherically convex hypersurfaces, also hold under the weaker condition of positive sectional curvature. In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and homogeneous degree one function $f$ of the shifted principal curvatures $lambda_i=kappa_i-1$, plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where $f$ is assumed to satisfy a certain condition on the second derivatives. We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, then the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology. As applications of the convergence result, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of Alexandrov-Fenchel type inequalities for smooth horospherically convex hypersurfaces in hyperbolic space.
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We consider the quermassintegral preserving flow of closed emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the princ
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The Alexandrov--Fenchel inequality bounds from below the square of the mixed volume $V(K_1,K_2,K_3,ldots,K_n)$ of convex bodies $K_1,ldots,K_n$ in $mathbb{R}^n$ by the product of the mixed volumes $V(K_1,K_1,K_3,ldots,K_n)$ and $V(K_2,K_2,K_3,ldots,K
_n)$. As a consequence, for integers $alpha_1,ldots,alpha_minmathbb{N}$ with $alpha_1+cdots+alpha_m=n$ the product $V_n(K_1)^{frac{alpha_1}{n}}cdots V_n(K_m)^{frac{alpha_m}{n}} $ of suitable powers of the volumes $V_n(K_i)$ of the convex bodies $K_i$, $i=1,ldots,m$, is a lower bound for the mixed volume $V(K_1[alpha_1],ldots,K_m[alpha_m])$, where $alpha_i$ is the multiplicity with which $K_i$ appears in the mixed volume. It has been conjectured by Ulrich Betke and Wolfgang Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume $V(K_1[alpha_1],ldots,K_m[alpha_m])$ in terms of the product of the intrinsic volumes $V_{alpha_i}(K_i)$, for $i=1,ldots,m$. The case where $m=2$, $alpha_1=1$, $alpha_2=n-1$ has recently been settled by the present authors (2020). The case where $m=3$, $alpha_1=alpha_2=1$, $alpha_3=n-2$ has been treated by Artstein-Avidan, Florentin, Ostrover (2014) under the assumption that $K_2$ is a zonoid and $K_3$ is the Euclidean unit ball. The case where $alpha_2=cdots=alpha_m=1$, $K_1$ is the unit ball and $K_2,ldots,K_m$ are zonoids has been considered by Hug, Schneider (2011). Here we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov--Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.
We consider the flow of closed convex hypersurfaces in Euclidean space $mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the m
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