HYPERBOLIC ALEXANDROV-FENCHEL QUERMASSINTEGRAL INEQUALITIES II

In this paper we first establish an optimal Sobolev-type inequality for hypersurfaces in H-n (see Theorem 1.1). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. More precisely, we prove a geometric inequality in the hyperbolic space H-n, which is a hyperbolic Alexandrov-Fenchel inequality, integral(Sigma)sigma(2k) >= C(n-1)(2k)wn-1{(vertical bar Sigma vertical bar/wn-1)(1/k) + (vertical bar Sigma vertical bar/w(n-1))(1/k n-1-2k/n-1)}(k), When Sigma is a horospherical convex and 2k <= n - 1. Equality holds if and only if Sigma is a geodesic sphere in H-n. Here sigma j = sigma(j)(k) is the jth mean curvature and k = (k(1), k(2), ... , k(n-1)) is the set of the principal curvatures of Sigma. Also, an optimal inequality for quermassintegral in H-n is W2k+1 (Omega) >= w(n-1)/n Sigma(k)(i=0) n-1-2k/n-1-2k+2i C-k(i) (nW(1)(Omega)/w(n-1)) (n-1-2k+2i/n-1), provided that Omega subset of H-n is a domain with Sigma = partial derivative Omega horospherical convex, where 2k <= 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.