We show that in the analytic category, given a Riemannian metric g on a hypersurface M ⊂ Z and a symmetric tensor W on M , the metric g can be locally extended to a Riemannian Einstein metric on Z with second fundamental form W , provided that g and W satisfy the constraints on M imposed by the contracted Gauss and CodazziMainardi equations. We use this fact to study the Cauchy problem for metr...

We study the nonlinear elliptic problem with discontinuous nonlinearity −∆u = f(u)H(u− μ) in Ω, u = h on ∂Ω, where H is the Heaviside unit function, f, h are given functions and μ is a positive real parameter. The domain Ω is the unit ball in Rn with n ≥ 3. We show the existence of a positive solution u and a hypersurface separating the region where −∆u = 0 from the region where −∆u = f(u). Our...

We prove that the k-truncated microsupport of the specialization of a complex of sheaves F along a submanifold is contained in the normal cone to the conormal bundle along the k-truncated microsupport of F . In the complex case, applying our estimates to F = RHomD(M,O), where M is a coherent D-module, we obtain new estimates for the truncated microsupport of real analytic solutions. When M is r...

We introduce new biholomorphic invariants for real-analytic hypersurfaces in C and show how they can be used to show that a hypersurface possesses few automorphisms. We give conditions, in terms of the new invariants, guaranteeing that the stability group is finite, and give (sharp) bounds on the cardinality of the stability group in this case. We also give a sufficient condition for the stabil...

In the previous letter, arXiv:2210.16738[hep-th], we found a set of flavor mass relations as constraints that $\beta$-deformed $A_{n-1}$ quiver matrix model restores maximal symmetry in massive scaling limit and reported existence Argyres-Douglas critical hypersurface. this derive concrete conditions on moduli parameters which maximally degenerates Seiberg-Witten curve while maintaining relatio...