Untitled

Abstract

Conformal deformations play an important role in global geometry. In general, such deformations are guided by certain partial differential equations. Yamabe problem is one of the examples. In this paper, we are interested in a class of fully nonlinear differential equations related to the deformation of conformal metrics.Let (M, g 0 ) be a compact connected smooth Riemannian manifold of dimension n ≥ 3, and let [g 0 ] denote the conformal class of g 0 . The Schouten tensor of the metric g is defined aswhere Ric g and R g are the Ricci tensor and scalar curvature of g, respectively. This tensor is connected to the study of conformal invariants, in particular, conformally invariant tensors and differential operators (see, e.g., [6] and the references therein). The following σ k -scalar curvatures of g were considered by Viaclovsky in [17]:where σ k is the kth elementary symmetric function and g −1 · S g is locally defined by (g −1 · S g ) i j = g ik (S g ) kj . When k = 1, σ 1 -scalar curvature is just the scalar curvature R (up

Existence and Uniqueness of Green's Functions to Nonlinear Yamabe Problems

“…x N / is uniformly bounded as t 3 1. Furthermore, as u t x u, known interior first derivative estimates [10,22], [37, theorem 1.10], [55], and interior second derivative estimates in [22] and [38, theorem 1.20] give…”

“…We start with some well-known facts. By local gradient and second-derivative estimates [10,22,37,38,55], we have (5.4) jr `ln u i .x/j Cd g .x; p i / `in B g .p I ; 3r 0 =4/ n fp i g for `h 1; 2: For x P R n and > 0, let C > 1 (independent of i) such that, after passing to a subsequence, u i .x/ ! 1 C u i .p i / 1 d g .x; p i / .n 2/ in fr 0 !…”

Local pointwise second derivative estimates for strong solutions to the $$\sigma _k$$-Yamabe equation on Euclidean domains

“…(1.2) and their counterparts on Riemannian manifolds were first studied by Viaclovsky in [63]. Since then, these equations have been addressed by various authors -for a partial list of references, see [1][2][3][8][9][10][11][12]14,[16][17][18][19]24,27,28,[31][32][33]35,[39][40][41]43,44,46,47,53,54,56,64,65] in the positive case and [13,23,25,29,30,42,45,55] in the negative case. When k = 1, these equations reduce to the original Yamabe equation.…”

“…A priori local first and second derivative estimates play an important role in the study of the σ k -Yamabe equation, and were established in the positive case by Chen [14], Guan and Wang [27], Jin, Li and Li [39], Li and Li [40], Li [43] and Wang [65]. In the negative case, an a priori (global) C 1 estimate is proven by Gursky and Viaclovsky [30], but it is unknown whether a priori C 2 estimates hold.…”

A class of Neumann problems arising in conformal geometry

“…where ϕ is a given smooth function on M. If τ = 1 = k and ϕ is constant, (1-1) is just the Yamabe problem, which has been solved by Yamabe, Trudinger, Aubin and Schoen (see [Lee and Parker 1987]). When τ = 1, k ≥ 2 and ϕ is constant, then (1-1) is called k-Yamabe problem, which has attracted enormous interest [Chang et al 2002; Ge and Wang 2006;Guan and Wang 2003a;2003b;Gursky and Viaclovsky 2007;Li and Li 2003;2005;Trudinger and Wang 2009;2010;Viaclovsky 2000], etc. There are many interesting works on the Yamabe problem and k-Yamabe problem on a manifold with boundary [Chen 2007;Escobar 1992b;1992a;Han and Li 1999;2000;He and Sheng 2011a;2011b;, etc.…”