Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Gursky-Streets equation and its application to the $sigma_k$ Yamabe problem
92
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of $sigma_2$ Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform $C^{1, 1}$ estimates for $2kleq n$. An important new ingredient is to show the concavity of the operator which holds for all $kleq n$. Our proof of the concavity heavily relies on Gardings theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform $C^{1, 1}$ emph{a priori estimates} for all the cases $ngeq 2k$. Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time. As an application, we prove that if $kgeq 3$ and $M^{2k}$ is conformally flat, any solution solution of $sigma_k$ Yamabe problem is conformal diffeomorphic to the round sphere $S^{2k}$.
rate research
Read More
We study the problem of prescribing $sigma_k$-curvature for a conformal metric on the standard sphere $mathbb{S}^n$ with $2 leq k < n/2$ and $n geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $beta in [2,n)$. We prove among other things the following statements. (1) When $beta>n-2k$, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When $ beta = n-2k$, there is an explicit positive constant $C(K)$ associated with $K$. If $C(K)>1$, the solution set is compact with a nonzero total degree counting and is therefore non-empty. If $C(K)<1$, the solution set is compact but the total degree counting is $0$, and the solution set is sometimes empty and sometimes non-empty. (3) When $frac{2}{n-2k}le beta < n-2k$, the solution set is compact, but the total degree counting is zero, and the solution set is sometimes empty and sometimes non-empty. (4) When $beta < frac{n-2k}{2}$, there exists $K$ for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of $beta$, there exists also some $K$ for which the solution set is empty.
We prove that any positive solution of the Yamabe equation on an asymptotically flat $n$-dimensional manifold of flatness order at least $frac{n-2}{2}$ and $nle 24$ must converge at infinity either to a fundamental solution of the Laplace operator on the Euclidean space or to a radial Fowler solution defined on the entire Euclidean space. The flatness order $frac{n-2}{2}$ is the minimal flatness order required to define ADM mass in general relativity; the dimension $24$ is the dividing dimension of the validity of compactness of solutions to the Yamabe problem. We also prove such alternatives for bounded solutions when $n>24$. We prove these results by establishing appropriate asymptotic behavior near an isolated singularity of solutions to the Yamabe equation when the metric has a flatness order of at least $frac{n-2}{2}$ at the singularity and $n<24$, also when $n>24$ and the solution grows no faster than the fundamental solution of the flat metric Laplacian at the singularity. These results extend earlier results of L. Caffarelli, B. Gidas and J. Spruck, also of N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, when the metric is conformally flat, and work of C.C. Chen and C. S. Lin when the scalar curvature is a non-constant function with appropriate flatness at the singular point, also work of F. Marques when the metric is not necessarily conformally flat but smooth, and the dimension of the manifold is three, four, or five, as well as recent similar results by the second and third authors in dimension six.
In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in $mathbb{R}^n$.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf for some classes of PDE on compact manifolds (like prescribed scalar cuvature). We also have an upper bound for the same product but on any Riemannian manifold not necessarily compact. An application of those result is an uniqueness solution for some PDE.
We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $mgeq3$ starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe flow.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
