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.
We study the topological basis problem under ZF, the Zermelo-Fraenkel axiomatic set theory without the Axiom of Choice. We prove that under AD+, the class of regular topologies on has a three element basis and the class of Hausdorff topologies on has a single-element basis. In particular, AD+ implies that the class of uncountable regular spaces has a four element basis.
We define floating bodies in the class of -dimensional ball-convex bodies. A right derivative of volume of these floating bodies leads to a surface area measure for ball-convex bodies which we call relative affine surface area. We show that this quantity is a rigid motion invariant, upper semicontinuous valuation.
Let be a simple-artinian ring with involution. This means that is isomorphic to a matrix ring over a ring that is either a skew field or the direct sum of a skew field and its opposite, and is given in terms of an involution of . We show that an arbitrary pseudo-quadratic module defined over can be obtained by a tensor product construction from a pseudo-quadratic module defined over and we apply this result to give a uniform description of arbitrary pseudo-maximal parabolic subgroups of arbitrary classical groups in terms of pseudo-quadratic modules.
We study an optimal partition problem on the sphere, where the cost functional is associated with the fractional -curvature in terms of the conformal fractional Laplacian on the sphere. By leveraging symmetries, we prove the existence of a symmetric minimal partition through a variational approach. A key ingredient in our analysis is a new Hölder regularity result for symmetric functions in a fractional Sobolev space on the sphere. As a byproduct, we establish the existence of infinitely many solutions to a nonlocal weakly coupled competitive system on the sphere that remain invariant under a group of conformal diffeomorphisms and we investigate the asymptotic behavior of least-energy solutions as the coupling parameters approach negative infinity.
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