Related papers: Optimal control for the conformal Laplacian obstac…

In this paper we demonstrate that under general conditions there exists a metric in the conformal class of an arbitrary metric on a smooth, closed Riemannian manifold of dimension greater than four such that the $Q$-curvature of the metric…

In this paper, we establish the existence of conformal deformations that uniformize fourth order curvature on 4-dimensional Riemannian manifolds with positive conformal invariants. Specifically, we prove that any closed, compact Riemannian…

Given a compact Riemannian manifold with umbilic boundary, the Yamabe boundary problem studies if there exist conformal scalar-flat metrics such that the boundary has constant mean curvature. In this paper we address to the stability of…

This paper addresses an open problem in the area of linear quadratic optimal control. We consider the regular, infinite-horizon, stability-modulo-a-subspace, indefinite linear quadratic problem under the assumption that the dynamics are…

This paper presents an overview of recent developments in the analysis of shapes such as curves and surfaces through Riemannian metrics. We show that several constructions of metrics on spaces of submanifolds can be unified through the…

For a simply connected closed Riemannian manifold with positive scalar curvature, we prove an upper diameter bound in terms of its scalar curvature integral, the Yamabe constant and the dimension of the manifold. When a manifold has a…

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…

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…

For a smooth compact Riemannian manifold with positive Yamabe invariant, positive Q curvature and dimension at least 5, we prove the existence of a conformal metric with constant Q curvature. Our approach is based on the study of extremal…

In 1992, motivated by Riemann mapping theorem, Escobar considered a version of Yamabe problem on manifolds of dimension n greater than 2 with boundary. The problem consists in finding a conformal metric such that the scalar curvature is…

We consider an optimal control problem where the state is governed by a free boundary problem called the two-phase membrane problem and the control appears in the coefficients of the characteristic function of the positivity and negativity…

The Yamabe Invariant of a smooth compact manifold is by definition the supremum of the scalar curvatures of unit-volume Yamabe metrics on the manifold. For an explicit infinite class of 4-manifolds, we show that this invariant is positive…

We prove that the Yamabe invariant of any simply connected smooth manifold of dimension n greater than four is non-negative. Equivalently that the infimum of the L^{n/2} norm of the scalar curvature, over the space of all Riemannian metrics…

In this paper, we are concerned with optimal control problems evolved on Riemannian manifolds, where the initial and final states satisfy some inequality and equality type constraints, and the control set is a separable metric space. We…

Controlling the shapes of surfaces provides a novel way to direct self-assembly of colloidal particles on those surfaces and may be useful for material design. This motivates the investigation of an optimal control problem for surface shape…

We consider the product of a compact Riemannian manifold without boundary and null scalar curvature with a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary. We use bifurcation…

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any…

We consider the problem of finding a metric in a given conformal class with prescribed non-positive scalar curvature and non-positive boundary mean curvature on an asymptotically Euclidean manifold with inner boundary. We obtain a necessary…

On any closed Riemannian manifold of dimension $n\geq 3$, we prove that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close, in a quantitative sense, to a minimizing Yamabe metric in the…

On a compact Riemannian manifold with boundary, we study the set of conformal metrics of negative constant scalar curvature in the interior and positive constant mean curvature on the boundary. Working in the case of positive Yamabe…