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Register a new userConformal metrics with prescribed scalar and mean curvature
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Authors
S. Cruz-Blazquez
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We consider the case with boundary of the classical Kazdan-Warner problem in dimension greater or equal than three, i.e. the prescription of scalar and boundary mean curvatures via conformal deformations of the metric. We deal in particular with negative scalar curvature and boundary mean curvature of arbitrary sign, which to our knowledge has not been treated in the literature. We employ a variational approach to prove new existence results, especially in three dimensions. One of the principal issues for this problem is to obtain compactness properties, due to the fact that bubbling may occur with profiles of hyperbolic balls or horospheres, and hence one may lose either pointwise estimates on the conformal factor or the total conformal volume. We can sometimes prevent them using integral estimates, Pohozaev identities and domain-variations of different types.
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We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n geq 3$, assume the conformal invariant $Y(M,partial M)<0$. Given any negative smooth functions $f$ in $M$ and $h$ on $partial M$, there exists a unique conformal metric of $g_0$ such that its scalar curvature equals $f$ and mean curvature curvature equals $h$. The first two methods are sub-super-solution method and subcritical approximation, and the third method is a geometric flow. In the flow approach, assume another conformal invariant $Q(M,pa M)$ is a negative real number, for some class of initial data, we prove the short time and long time existences of the so-called prescribed scalar curvature plus mean curvature flows, as well as their asymptotic convergence. Via a family of such flows together with some additional variational arguments, under the flow assumptions we prove existence and uniqueness of positive minimizers of the associated energy functional and also the above result by analyzing asymptotic limits of the flows and the relations among some conformal invariants.
We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension $ngeq 3$. We prove the existence of such conformal metrics in the cases of $n=6,7$ or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be $1$, there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to $+infty$.
We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.
We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the link of the singular set. Within this class of conic metrics, we determine obstructions to the existence of conformal deformations to constant scalar curvature of any sign (positive, negative, or zero). For conic metrics with negative scalar curvature, we determine sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1; moreover, we show that this metric is unique within its conformal class of conic metrics. Our work is in dimensions three and higher.
We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of fully nonlinear equations. Our method is appropriate and delicate as shown by a topological obstruction. Finally, we discuss Hessian equations and Weingarten equations by confirming a key assumption.
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