Some Dirichlet problems arising from conformal geometry

Abstract

We study the problem of finding complete conformal metrics determined by some symmetric function of the modified Schouten tensor on compact manifolds with boundary; which reduces to a Dirichlet problem. We prove the existence of the solution under some suitable conditions. In particular, we prove that every smooth compact n-dimensional manifold with boundary, with n ≥ 3, admits a complete Riemannian metric g whose Ricci curvature Ric g and scalar curvature R g satisfyThis result generalizes Aviles and McOwen's in the scalar curvature case.

A class of Neumann problems arising in conformal geometry

“…We first sketch the ellipticity properties of operator F; see [Li and Sheng 2011] for details. For any function h on M, we define…”

“…We use the continuity method to prove the existence of (1-6). Since the argument is standard (see [Li and Sheng 2011]), we only sketch it here. For t ∈ [0, 1], consider the equation…”

“…By choosing l = 0 and ϕ constant in Corollaries 1.4 and 1.5, we can get Theorem 1.1 directly. Different from the results in [Li and Sheng 2011;, we need not subjoin any restriction on a(x) and b(x) in Theorems 1.2 and 1.3. Contrary to this fact, gives a counterexample to show that there is no regularity if a(x) = 0 and b(x) > 0 when τ = 1 and…”

“…Using a similar argument, Sheng and Zhang [2007] studied the case of τ > n − 1, A τ g ∈ + k and ϕ > 0. For the manifold with boundary, Li and Sheng [2011] considered a Dirichlet problem of (1-1) for τ > n − 1 and A τ g ∈ + k ; He and Sheng [2013] discussed more general equations and obtained many useful local estimates for both τ < 1 and τ > n − 1. In [Sheng and Yuan 2013], we investigated a Neumann problem of (1-1) by a conformal flow and proved: Theorem 1.1 [Sheng and Yuan 2013].…”

The Dirichlet problem for a class of Hessian quotient equations on Riemannian manifolds

“…where ν was the unit outer normal vector of M. Later on, The authors [2] studied the corresponding Hessian quotient type prescribed curvature problem. Moreover, an analogue of equation (1.1) on compact manifolds also appeared naturally in conformal geometry, see Gursky-Viaclovsky [14], Li-Sheng [23] and Sheng-Zhang [29].…”