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The topological censorship theorem suggests that higher dimensional black holes can possess the domain of outer communication (DOC) of nontrivial topology. In this paper, we seek for a black hole coexisting with two bubbles adjacent to the…
We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of $6$-dimensional (stable minimal) immersions. The key step is…
We present new quantitative estimates for the radially symmetric configuration concerning Serrin's overdetermined problem for the torsional rigidity, Alexandrov's Soap Bubble Theorem, and other related problems. The new estimates improve on…
A classical result owing to Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] asserts that all positive solutions of the Poincar\'e-Sobolev equation on the hyperbolic space $$ -\Delta_{\mathbb{B}^n} u-\lambda u =…
A flexible polyhedron in an n-dimensional space of constant curvature, namely, in the Euclidean space, or in the Lobachevsky space, or in the sphere, is a polyhedron with rigid (n-1)-dimensional faces and hinges at (n-2)-dimensional faces.…
Complex scalar fields charged under a global U(1) symmetry can admit non-topological soliton configurations called Q-balls which are stable against decay into individual particles or smaller Q-balls. These Q-balls are interesting objects…
In this paper, we present a new explicit formula for the curvatures of a regular curve with an arbitrary parameter in the Euclidean space $\mathbb{R}^n$, $n\geq 2$, expressed only in terms of its derivatives. We introduce also the notion of…
Through the application of layer potential techniques and Gohberg-Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole…
For each $d\geq 3$ we construct cube complexes homeomorphic to the $d$-sphere with $n$ vertices in which the number of facets (assuming $d$ constant) is $\Omega(n^{5/4})$. This disproves a conjecture of Kalai's stating that the number of…
We use branes to generalize the Distance Conjecture. We conjecture that in any infinite-distance limit in the moduli space of a $d$-dimensional quantum gravity theory, among the set of particle towers and fundamental branes with at most…
We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good…
Given $ n \geq 2 $ and $ k \in \{2, \ldots , n\} $, we study the asymptotic behaviour of sequences of bounded $C^2$-domains of finite total curvature in $ \mathbb{R}^{n+1} $ converging in volume and perimeter, and with the $ k $-th mean…
We investigate the behaviour of the spectrum of the quantum (or Dubrovin) connection of smooth projective surfaces under blow-ups. Our main result is that for small values of the parameters, the quantum spectrum of such a surface is…
This note presents two nontrivial, rotational equilibrium solutions to the spatial uniform gas pressure (isobaric) approximate model of Prosperetti in the inviscid case. Building on Gavrilov's work [GAFA 2019], we first establish the…
We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete…
We present a class of spherically symmetric spacetimes corresponding to bubbles separating two regions with constant values of the scalar curvature, or equivalently with two different cosmological constants, in quadratic F(R) theory. The…
The paper is concerned with the slightly subcritical elliptic problem with Hardy term \[ \left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &= |u|^{2^{\ast}-2-\epsilon}u &&\quad \text{in } \Omega, \\\ u &= 0&&\quad \text{on }…
In this work, we generalize Sacks-Uhlenbeck's existence result for harmonic spheres, constructing for $n \ge 2$, regular, non-trivial, $n$-harmonic $n$-spheres into suitable target manifolds. We obtain an infinite family of new…
The different definitions for the equation of state of a non-minimally coupled scalar field that have been introduced in the literature are analyzed. Particular emphasis is made upon those features that could yield to an observable way of…
Theories with compact extra dimensions are sometimes unstable to decay into a bubble of nothing -- an instability resulting in the destruction of spacetime. We investigate the existence of these bubbles in theories where the moduli fields…