Related papers: The Eigencurve is Proper

A curve $\gamma$ in a Riemannian manifold $M$ is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when $\gamma$ lies on an oriented…

We study solutions of a Euclidean weighted porous medium equation when the weight behaves at spacial infinity like $|x|^{-\gamma}$, for $\gamma\in [0,2)$, and is allowed to be singular at the origin. In particular we show local-in-time…

Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. We give a complete classification of when…

We prove a Heinz type inequality for harmonic diffeomorphisms of of the half-plane onto itself. We then apply this result to prove some sharp bound of the Gaussian curvature of a minimal surface, provided that it lies above the whole…

We prove that in a stable range, the rational cohomology of the moduli space of curves with level structures is the same as that of the ordinary moduli space of curves.

We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $\Sigma$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation…

We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the…

We prove that every endomorphism of the mapping class group of an orientable surface onto a subgroup of finite index is in fact an automorphism.

Fix an elliptic curve E over Q. An extremal prime for E is a prime p of good reduction such that the number of rational points on E modulo p is maximal or minimal in relation to the Hasse bound. Assuming that all the symmetric power…

The curvature tensor and the scalar curvature are computed in the space of positive definite real matrices endowed by the Kubo-Mori inner product as a Riemannian metric.

We show that the use of the main characteristics of the circle map leads naturally to establish a few statements on primes and pseudoprimes. In this way a Fermat's theorem on primes and some interesting properties of pseudoprimes are…

Building on work of Farb and the second author, we prove that the group of automorphisms of the fine curve graph for a surface is isomorphic to the group of homeomorphisms of the surface. This theorem is analogous to the seminal result of…

We obtain a Chern-Osserman type equality of a complete properly immersed surface in Euclidean space, provided the L^2-norm of the second fundamental form is finite. Also, by using a monotonicity formula, we prove that if the L^2-norm of…

We give a simple proof of Voisin's Theorem for general canonical curves of even genus. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves.

We prove Obata's rigidity theorem for metric measure spaces that satisfy a Riemannian curvature-dimension condition. Additionally, we show that a lower bound $K$ for the generalized Hessian of a sufficiently regular function $u$ holds if…

Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $\partial M$ and…

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a $C^{1,\lambda}$-a-priori bound for surfaces for which this functional is finite. In fact, it turns out…

Given a prime $p$ and cusp forms $f_1$ and $f_2$ on some $\Gamma_1(N)$ that are eigenforms outside $Np$ and have coefficients in the ring of integers of some number field $K$, we consider the problem of deciding whether $f_1$ and $f_2$ have…

We give a proof of the celebrated stability theorem of Perelman stating that for a noncollapsing sequence $X_i$ of Alexandrov spaces with curvature bounded below Gromov-Hausdorff converging to a compact Alexandrov space $X$, $X_i$ is…

We prove the existence of embedded closed constant curvature curves on convex surfaces.