Let M be a compact Riemannian manifold of dimension n. The k-curvature, for k = 1, 2, · · · , n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a conformal metric whose k-curvature is a constant. When k = 1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions to the k-Yamabe problem was recently proved by Gursky and Viaclovsky for k > n 2 . In this paper we prove the existence of solutions for the remaining cases 2 ≤ k ≤ n 2 , assuming that the equation is variational.
In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space R n+1 with positive mean curvature is κ-noncollapsing, and a blow-up sequence converges locally smoothly along a subsequence to a smooth, convex blow-up solution. As a consequence we obtain a local Harnack inequality for the mean convex flow.
In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space R n+1 with speed f r α K, where K is the Gauss curvature, r is the distance from the hypersurface to the origin, and f is a positive and smooth function. If α ≥ n + 1, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if f ≡ 1. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang [29], for the case q < 0. If α < n+1, corresponding to the case q > 0, we also establish the same results for even function f and originsymmetric initial condition, but for non-symmetric f , counterexample is given for the above smooth convergence.2010 Mathematics Subject Classification. 35K96, 53C44.
We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Monge-Ampère type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C 1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.
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