We establish conditions for nonexistence of global solutions for a class of quasilinear parabolic problems with a potential on complete, non-compact Riemannian manifolds, including the Porous Medium Equation and the p-Laplacian with a potential term. Our results reveal the interplay between the manifold's geometry, the power nonlinearity, and the potential's behavior at infinity. Using a test function argument, we identify explicit parameter ranges where nonexistence holds.
We give some criteria for the Lie algebra to be solvable, where is a -block of a finite group algebra, in terms of the action of an inertial quotient of on a defect group of .
Let be the moduli space of hyperbolic surfaces of genus endowed with the Weil–Petersson metric. We view the regularized determinant of Laplacian as a function on and show that there exists a universal constant such that as ,
We study local–global principles for semi-integral points on orbifold pairs of Markoff type. In particular, we analyse when these orbifold pairs satisfy weak weak approximation, weak approximation and strong approximation off a finite set of places. We show that Markoff orbifold pairs satisfy the semi-integral Hasse principle and we measure how often such orbifold pairs have strict semi-integral points but the corresponding Markoff surface lacks integral points.
Tambara and Yamagami investigated a simple set of fusion rules with only one noninvertible object, and proved under which circumstances those rules could be given a coherent associator. They also classified all of the resulting fusion categories up to monoidal equivalence. We consider a generalization of such fusion rules to the setting where simple objects are no longer required to be split simple. Over the real numbers, this means that simple objects are either real, complex, or quaternionic. In this context, we prove a similar categorification result to the one of Tambara and Yamagami.
Given a conformal action of a discrete group on a Riemann surface, we study the maximization of Laplace and Steklov eigenvalues within a conformal class, considering metrics invariant under the group action. We establish natural conditions for the existence and regularity of maximizers. Our method simplifies the previously known techniques for proving existence and regularity results in conformal class optimization. Finally, we provide a complete solution to the equivariant maximization problem for Laplace eigenvalues on the sphere and Steklov eigenvalues on the disk, resolving open questions posed by Arias-Marco et al. (2024) regarding the sharpness of the Hersch–Payne–Schiffer inequality and the maximization of Steklov eigenvalues by the standard disk among planar simply connected domains with symmetry.
In this paper, we consider the compactness of the solutions to the equivariant Yamabe equation on manifolds with boundary. We construct a smooth counterexample showing that the compactness of the set of “lower energy” solutions to the equivariant Yamabe equation fails when the dimension of the manifold is not less than 25.
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