Abstract. Given a positive function F on S n which satisfies a convexity condition, for 1 ≤ r ≤ n, we define the r-th anisotropic mean curvature function H F r for hypersurfaces in R n+1 which is a generalization of the usual r-th mean curvature function. We prove that a compact embedded hypersurface without boundary in R n+1 with H F r = constant is the Wulff shape, up to translations and homotheties. In case r = 1, our result is the anisotropic version of Alexandrov Theorem, which gives an affirmative answer to an open problem of F. Morgan.
We give a proof of the DDVV conjecture, which is a pointwise inequality involving the scalar curvature, the normal scalar curvature and the mean curvature on a submanifold of a real space form. We also solve the problem of its equality case.
Dedicated to Professor Banghe Li on his 70th birthday.Abstract. The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the GromollMeyer sphere. In particular, as one of main results we prove there exists no properly transnormal function on any exotic 4-sphere if it exists. Furthermore, by projecting an S 3 -invariant isoparametric function on Sp(2), we construct a properly transnormalbut not an isoparametric function on the Gromoll-Meyer sphere with two points as the focal varieties.
In our previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exists at least one minimal isoparametric hypersurface. In this paper, we show such a minimal isoparametric hypersurface is also unique in the family if the ambient manifold has positive Ricci curvature. Moreover, we give a proof of Theorem D claimed by Q.M.Wang (without proof) which asserts that the focal submanifolds of an isoparametric function on a complete Riemannian manifold are minimal. Further, we study isoparametric hypersurfaces with constant principal curvatures in general Riemannian manifolds. It turns out that in this case the focal submanifolds have the same properties as those in the standard sphere, i.e., the shape operator with respect to any normal direction has common constant principal curvatures. Some necessary conditions involving Ricci curvature and scalar curvature are also derived.2010 Mathematics Subject Classification. 53C20.
We compute the cohomology of crystallographic groups Γ = Z n Z/p with holonomy of prime order by establishing the collapse at E 2 of the spectral sequence associated to their defining extension. As an application we compute the group of gerbes associated to many six-dimensional toroidal orbifolds arising in string theory.
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