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Article overview
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Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II | Yuxin Ge
; Guofang Wang
; Jie Wu
; | Date: |
4 Apr 2013 | Abstract: | In this paper we first establish an optimal Sobolev type inequality for
hypersurfaces in $H^n$(see Theorem
ef{mainthm1}). As an application we
obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and
quermassintegrals. Precisely, we prove a following geometric inequality in the
hyperbolic space $H^n$, which is a hyperbolic Alexandrov-Fenchel inequality,
egin{equation*} egin{array}{rcl} ds int_Sigma s_{2k}ge dsvs
C_{n-1}^{2k}omega_{n-1}left{left(frac{|Sigma|}{omega_{n-1}}
ight)^frac 1k + left(frac{|Sigma|}{omega_{n-1}}
ight)^{frac 1kfrac
{n-1-2k}{n-1}}
ight}^k, end{array} end{equation*} provided that $Sigma$
is a horospherical convex, where $2kleq n-1$. Equality holds if and only if
$Sigma$ is a geodesic sphere in $H^n$. Here $sigma_{j}=s_{j}(kappa)$ is
the $j$-th mean curvature and $kappa=(kappa_1,kappa_2,cdots, kappa_{n-1})$
is the set of the principal curvatures of $Sigma$. Also, an optimal inequality
for quermassintegrals in $H^n$ is as following: $$ W_{2k+1}(Omega)geqfrac
{omega_{n-1}}{n}sum_{i=0}^kfrac{n-1-2k}{n-1-2k+2i},C_k^iigg(frac{nW_1(Omega)}{omega_{n-1}}igg)^{frac{n-1-2k+2i}{n-1}},
$$ provided that $OmegasubsetH^n$ is a domain with $Sigma=partialOmega$
horospherical convex, where $2kleq n-1$. Equality holds if and only if
$Sigma$ is a geodesic sphere in $H^n$. Here $W_r(Omega)$ is
quermassintegrals in integral geometry. | Source: | arXiv, 1304.1417 | Services: | Forum | Review | PDF | Favorites |
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