相关论文: The Hijazi inequality on conformally parabolic man…

A rigorous \textit{ab initio} derivation of the (square of) Dirac's equation for a single particle with spin is presented. The general Hamilton-Jacobi equation for the particle expressed in terms of a background Weyl's conformal geometry is…

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…

In K\"ahler-Einstein case of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which…

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the boundary. In the situations we consider, we…

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove…

We are concerned with the dependence of the lowest positive eigenvalue of the Dirac operator on the geometry of rectangles, subject to infinite-mass boundary conditions. We conjecture that the square is a global minimiser both under the…

We will present an estimate for the first eigenvalue of the Dirichlet and Neumann problems in terms of the Bakry-\'Emery Ricci curvature for a compact weighted manifold. As an application we will establish a stability condition for a…

We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S^3 we give a simple criterion in terms of linking…

We show that the Dirac operator on a spin manifold does not admit $L^2$ eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if…

In the paper, we consider a path-dependent Hamilton-Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We…

We prove matrix and scalar differential Harnack inequalities for linear parabolic equations on Riemannian and K\"ahler manifolds.

We study here compact manifolds with positive scalar curvature metrics. We use the relative Yamabe invariant from math.DG/0008138 to define the conformal cobordism relation on the category of such manifolds. We prove that corresponding…

In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical…

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…

The conditions for R-separation of variables for the conformally invariant Laplace equation on an n-dimensional Riemannian manifold are determined and compared with the conditions for the additive separation of the null geodesic…

For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained holographically in a construction that uses a singular Yamabe…

We study the set of volumes of constant scalar curvature one metrics on an atoroidal three-manifold.The infinum of this set is believed to be attained at a hyperbolic metric. We prove that the supremum of this set is always infinity. The…

In this paper we construct and study isoperimetric functions at infinity for Hadamard manifolds. These quasi-isometry invariants give a measure of the spread of geodesics in such a manifold.

We study the filling invariants at infinity div_{k} for Hadamard manifolds defined by Brady and Farb in [Filling-invariants at infinity for manifolds of nonpositive curvature]. Among other results, we give a positive answer to the question…

For non-trivial solutions to the zero mode equation on a closed spin manifold \[D \varphi=iA\cdot \varphi,\] we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$…