In this paper, we prove a generalization of Reilly's formula in [10]. We apply such general Reilly's formula to give alternative proofs of the Alexandrov's Theorem and the Heintze-Karcher inequality in the hemisphere and in the hyperbolic space. Moreover, we use the general Reilly's formula to prove a new Heintze-Karcher inequality for Riemannian manifolds with boundary and sectional curvature bounded below.

In this paper, we prove the existence of a classical solution to a Neumann boundary problem for Hessian equations in uniformly convex domain. The methods depend upon the established of a priori derivative estimates up to second order. So we give a affirmative answer to a conjecture of N. Trudinger in 1986.First for the Hessian equation on R n , its Dirichlet boundary value problemwas studied by Caffrelli-Nirenberg-Spruck [2], Ivochkina [19] and Trudinger [39]. Chou-Wang [7] got the Pogorelov type interior estimates and the existence of variational solution. Trudinger-Wang [40] developed a Hessian measure theory for Hessian operator. For the curvature equations in classical geometry, the existence of hypersurfaces with prescribed Weingarten curvature was studied by Pogorelov [34], Caffarelli-Nirenberg-Spruck [3, 4], Guan-Guan [14], Guan-Ma [15] and the later work by Sheng-Trudinger-Wang [35]. The Hessian equation on Riemannian manifolds was also studied by Y.Y.Li [22], Urbas [44] and Guan [13]. In recent years the Hessian type equation also appears in conformal geometry, which started from Chang-Gursky-Yang [5] and the related development by ([16], [23], [17], [36],[10]). In Kähler geometry, the Hessian equation was studied by Hou-Ma-Wu [18] and Dinew-Kolodziej [8].The Yamabe problem on manifolds with boundary was first studied by Escobar [9], he shows that (almost) every compact Riemannian manifold (M, g) is conformally equivalent to one of constant scalar curvature, whose boundary is minimal. The problem reduces to solving the semilinear elliptic critical Sobolev exponent equation with the Neumann boundary condition. It is naturally, the Neumann boundary value problem for Hessian type equations also appears in the fully nonlinear Yamabe problem for manifolds with boundary, which is to find a conformal metricĝ = exp(−2u)g such that the k-th elementary symmetric function of eigenvalues of Schouten tensor is constant and with the constant mean curvature on the boundary of manifold. See for Jin-Li-Li [21], Chen [6] and Li-Luc [24,25], but in all these papers they need to impose the manifold are umbilic or total geodesic boundary for k ≥ 2, which are more like the condition in Trudinger [38] that the domain is ball.The Neumann or oblique derivative problem on linear and quasilinear elliptic equations was widely studied for a long time, one can see the recent book written by Lieberman [27]. Especially for the mean curvature equation with prescribed contact angle boundary value problem, Ural'tseva [41], and Gerhardt [11] got the boundary gradient estimates and the corresponding existence theorem. Recently in [33], Ma-Xu got the boundary gradient estimates and the corresponding existence theorem for the Neumann boundary value problem on mean curvature equation. For related results on the Neumann or oblique derivative problem for some class fully nonlinear elliptic equations can be found in Urbas [42,43].We give a brief description of our procedures and ideas to this problem. By the standard theory of Lieberman-Trudinge...

We establish interior C 2 estimates for convex solutions of scalar curvature equation and σ2-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces (M n , g) ⊂ R n+1 with positive scalar curvature. These estimates are consequences of an interior estimates for these equations obtained under a weakened condition.

Recently, the first named author together with Xinan Ma [11], have proved the existence of the Neumann problems for Hessian equations. In this paper, we proceed further to study classical Neumann problems for Hessian equations. We prove here the existence of classical Neumann problems under the uniformly convex domain in R n . As an application, we use the solution of the classical Neumann problem to give a new proof of a family of Alexandrov-Fenchel inequalities arising from convex geometry. This geometric application is motivated from Reilly [17].