Related papers: Singular Yamabe metrics by equivariant reduction
We consider the problem of constructing solutions to the Yamabe equation (i.e. conformal constant scalar curvature metrics) on the generalized connected sum M = (M_1) #_K (M_2) of two compact Riemannian manifolds (M_1,g_1) and (M_2,g_2)…
One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $\sigma_k$-curvature in the interior and constant…
The negative case of the Singular Yamabe Problem concerns the existence and behavior of complete metrics with constant negative scalar curvature on the complement of a closed set in a compact Riemannian manifold which are conformally…
We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type. Our first existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic. In…
We study Yamabe-type equations on the product of two spheres $(S^n \times S^n, G_\delta)$, where $G_\delta$ is a family of Riemannian metrics parametrized by $\delta > 0$. Using bifurcation theory and isoparametric functions, we establish…
The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the constant mixed scalar curvature for a…
We develop the calculus for hypersurface variations based on variation of the hypersurface defining function. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the…
We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…
We prove nonuniqueness results for complete metrics with constant positive fractional curvature conformal to the round metric on $S^n \setminus S^k$, using bifurcation techniques. These are singular (positive) solutions to a non-local…
We prove existence results for nodal solutions of the Yamabe equation that are constant along the level sets of an isoparametric function.
We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar…
We investigate the singular sets of solutions of conformally covariant elliptic operators of fractional order with the goal of developing generalizations of some well-known properties of solutions of the singular Yamabe problem.
In this paper, we investigate the existence of sign-changing solutions to a critical elliptic equation involving a Yamabe type operator on a compact manifold with boundary. The existence result is assured under some geometric conditions.
Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant…
In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations…
In this paper, we study the uniqueness of type II Yamabe metrics in conformal classes on a compact connected manifold with boundary, and we investigate Obata-type theorems for type II Yamabe metrics. In particular, we establish a theorem…
In this paper, we investigate the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive optimal estimates, where the background metrics are not assumed…
We consider a nonlinear version of the Yamabe problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary $C^2$ estimates directly from boundary $C^0$ estimates. In particular, the…
We compute the Yamabe invariant for a class of symplectic 4-manifolds of general type obtained by taking the rational blowdown of Kahler surfaces. In particular, for any point on the half-Noether line we exhibit a simply connected minimal…
The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding…