A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren–Poon monotonicity formula combined with a blow-up analysis.
A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren–Poon monotonicity formula combined with a blow-up analysis.
Let be a general Brill–Noether curve. A classical problem is to determine when , which controls the quadric section of .
So far this problem has only been solved in characteristic zero, in which case with finitely many exceptions. In this paper, we extend these results to positive characteristic, uncovering a wealth of new exceptions in characteristic .
For the largest exceptional simple Lie superalgebra , having dimension , we provide two explicit geometric realizations as supersymmetries, namely as the symmetry superalgebra of super-PDE systems of second- and third-order, respectively.
We study Kähler metrics on the total space of Griffiths negative holomorphic vector bundles over Kähler manifolds. As an application, we construct mapping class group invariant Kähler metrics on , the holomorphic tangent bundle of Teichmüller space of a closed surface . Consequently,we obtain a new mapping class group invariant Kähler metric on the quasi-Fuchsian space , which extends the Weil–Petersson metric on the Teichmüller space . We also calculate its curvature and prove non-positivity for the curvature along the tautological directions.
We study the relation between two special classes of Riemannian Lie groups with a left-invariant metric : The Einstein Lie groups, defined by the condition , and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Nikonorov. Our approach involves studying and characterizing the -invariant geodesic orbit metrics on Lie groups for a wide class of subgroups that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds.
In this paper, we prove a comparison principle for sub-supersolutions to a singular quasilinear problem that involves the anisotropic Finsler operator
We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework.
We prove that the metric tensor of a complete Riemannian manifold is uniquely determined, up to isometry, from the knowledge of a local source-to-solution operator associated with a fractional power of the Laplace–Beltrami operator . Our result holds under the condition that the metric tensor is known in an arbitrary small subdomain. We also consider the case of closed manifolds and provide an improvement of the main result in [A. Feizmohammadi, T. Ghosh, K. Krupchyk and G. Uhlmann, Fractional anisotropic Calderón problem on closed Riemannian manifolds, preprint (2021); arXiv:2112.03480].
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