The Yamabe problem for higher order curvatures

By Weimin Sheng, Neil S. Trudinger, Xu-jia Wang. 2005

182 citations recorded by the source corpus.

Let M be a compact Riemannian manifold of dimension n > 2. 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 is known for the case k = 2, n = 4, for locally conformally flat manifolds and for the cases k > n/2. In this paper we prove the solvability of the k-Yamabe problem in the remaining cases k ≤ n/2, under the hypothesis that the problem is variational. This includes all of the cases k = 2 as well as the locally conformally flat case.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/74aa19e7072217565275c26e47333ea4454e8c95

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Local estimates for some fully nonlinear elliptic equations (2005, background): “linear equations as described above, the coeﬃcient in front of the gradient square term plays an important role. For arbitrary coeﬃcients, in general it is not true that we have local estimates . See [14] for a counterexample. In the degenerate case, when the gradient square term disappears, one can”
The k-Hessian Equation (2009, unknown): “These results and techniques can also be used in other problems. See e.g., [FZ, KT, STW ].”
Local gradient estimates of solutions to some conformally invariant fully nonlinear equations (2006, unknown): “Such estimates were also studied in [12, 9, 19].”
SECOND-ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS (2012, background): “[20], [29], and in conformal geometry in which there is a huge literature; see for instance [6], [7], [8], [12], [30], [31], [32], [33], [34], [40], [41], [42], [48], [53], [54] and references therein.”
The Neumann Problem for Hessian Equations (2015, methodology): “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 Kahler geometry, the Hessian equation was studied by Hou-Ma-Wu [18] and Dinew-Kolodzi”
On Harnack inequalities and singularities of admissible metrics in the Yamabe problem (2005, methodology): “(1.10) is a fully nonlinear partial difierential equation, which is elliptic if the eigenvalues ‚(Ag) 2 ik. Therefore to study problem (1.9), we always assume [g0]k 6= ;. Under this assumption, the k-Yamabe problem has been solved in [ 22 ] if 2 • k • n 2 and (1.9) is”
Estimates and existence results for a fully nonlinear Yamabe problem on manifolds with boundary (2006, background): “In [45] Viaclovsky established the existence result for (1.8) with (f,Γ) = (σ 1/n n ,Γn) for a class of manifolds.”
A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound (2012, methodology): “Extensions, as well as developments of new methods, have been made by Guan and Wang [19], Li and Li [26, 27], Gursky and Viaclovsky [21, 22], Ge and Wang [17], Sheng, Trudinger and Wang [38], Trudinger and Wang [40, 41], among others.”
Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds (2014, background): “= ψ(x)e (where Sg is the Schouten tensor of (M , g)) introduced by Viaclovsky [50] is connected with a natural fully nonlinear version of the Yamabe problem and has very interesting applications in conformal geometry; see for instance [5], [10], [26], [28], [35], [43] and references therein.”