Journal of Geometric Analysis最新文献

We study the geometry of m-regular domains within the Caffarelli-Nirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every m-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure.

We deal with Morrey spaces on bounded domains $\Omega$ obtained by different approaches. In particular, we consider three settings $M_{u,p}\left(\Omega \right)$ , $M_{u,p}\left(\Omega \right)$ and $M_{u,p}\left(\Omega \right)$ , where $0<p\le u<\infty$ , commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes $E_G\left(M_{u,p}\left(\Omega \right)\right)$ as well as $E_G\left(M_{u,p}\left(\Omega \right)\right)$ , and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether $\frac{n}{u}\ge \frac{1}{p}$ or $\frac{n}{u}<\frac{1}{p}$ plays a decisive role when it comes to the behaviour of these spaces.

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with corresponding different metrics.

We study the asymptotic behaviour of the partial density function associated to sections of a positive hermitian line bundle that vanish to a particular order along a fixed divisor Y. Assuming the data in question is invariant under an $S^1$ -action (locally around Y) we prove that this density function has a distributional asymptotic expansion that is in fact smooth upon passing to a suitable real blow-up. Moreover we recover the existence of the "forbidden region" R on which the density function is exponentially small, and prove that it has an "error-function" behaviour across the boundary $\partial R$ . As an illustrative application, we use this to study a certain natural function that can be associated to a divisor in a Kähler manifold.

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature bounded below and diameter bounded above, Gromov-Hausdorff convergence agrees with intrinsic flat convergence. In particular, the limiting current is essentially unique, has multiplicity one, and mass equal to the Hausdorff measure. Moreover, the limit spaces satisfy a constancy theorem.

We investigate problems related with the existence of square integrable holomorphic functions on (unbounded) balanced domains. In particular, we solve the problem of Wiegerinck for balanced domains in dimension two. We also give a description of $L_h^2$ -domains of holomorphy in the class of balanced domains and present a purely algebraic criterion for homogeneous polynomials to be square integrable in a pseudoconvex balanced domain in $C^2$ . This allows easily to decide which pseudoconvex balanced domain in $C^2$ has a positive Bergman kernel and which admits the Bergman metric.

We prove that $\frac{2}{d},\frac{2d-3}{{(d-1)}^2},\frac{2d-1}{d(d-1)},\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest log canonical thresholds of reduced plane curves of degree $d⩾3$ , and we describe reduced plane curves of degree d whose log canonical thresholds are these numbers. As an application, we prove that $\frac{2}{d},\frac{2d-3}{{(d-1)}^2},\frac{2d-1}{d(d-1)},\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest values of the $\alpha$ -invariant of Tian of smooth surfaces in $P^3$ of degree $d⩾3$ . We also prove that every reduced plane curve of degree $d⩾4$ whose log canonical threshold is smaller than $\frac{5}{2d}$ is GIT-unstable for the action of the group ${\mathrm{PGL}}_3\left(C\right)$ , and we describe GIT-semistable reduced plane curves with log canonical thresholds  $\frac{5}{2d}$ .

This paper is devoted to the study of the medial axes of sets definable in polynomially bounded o-minimal structures, i.e. the sets of points with more than one closest point with respect to the Euclidean distance. Our point of view is that of singularity theory. While trying to make the paper self-contained, we gather here also a large bunch of basic results. Our main interest, however, goes to the characterization of those singular points of a definable, closed set $X\subset R^n$ , which are reached by the medial axis.