Related papers: A note on Seshadri constants on general $K3$ surfa…

We deal with the minimal Lagrangian surfaces of the Einstein-K\"ahler surface $S^2 \times S^2$, studying local geometric properties and showing that they can be locally described as Gauss maps of minimal surfaces in $S^3 \subset R^4$. We…

In this paper we generalize a weak sequential result of \cite{fan20182} to any non-scattering solutions in one dimension. No symmetry assumptions are required for the initial data.

In this paper we continue the study of algebraic fundamentale group of minimal surfaces of general type S satisfying K_S^2<3\chi(S). We show that, if K_S^2= 3\chi(S)-1 and the algebraic fundamental group of S has order 8, then S is a…

Let $\mathfrak{q}>2$ be a prime number, $\chi$ a primitive Dirichlet character modulo $\mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $\mathfrak{q}$ and trivial nebentypus. We prove the subconvex…

Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\mathbb Z)$, and $u$ an even Hecke-Maass form for $SL(2,\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound $O\big(k^{4/3+\varepsilon}\big)$ in the…

Suppose D is an effective divisor on a smooth projective algebraic variety X. For each point x of X we associate a numberical invariant called the moving Seshadri constant of D at x which is a numerical measure of positivity of the divisor…

We show that for any weakly convergent sequence of ergodic $SL_2(\mathbb{R})$-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of…

Let $L$ be a finite-dimensional real normed space, and let $B$ be the unit ball in $L$. The sign sequence constant of $L$ is the least $t>0$ such that, for each sequence $v_1, \ldots, v_n \in B$, there are signs $\varepsilon_1, \ldots,…

In this paper, we investigate eigenvalues of Laplacian on a bounded domain in an $n$-dimensional Euclidean space and obtain a sharper lower bound for the sum of its eigenvalues, which gives an improvement of results due to A. D. Melas [15].…

In this paper, we investigate the Seshadri constant $\varepsilon(X,T_X;p)$ of the tangent sheaf $T_X$ on a complete $\mathbb Q$-factorial toric variety $X$. We show that $\varepsilon(X,T_X;1)>0$ if and only if the following statement holds…

Let $M$ be a compact constant mean curvature surface either in $\mathbb{S}^3$ or $\mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded below by a linear function of the genus. As a by product we obtain a…

In this paper, we prove uniform bounds for $\rm GL (3)\times GL(2)$ $L$-functions in the $\rm GL(2)$ spectral aspect and the $t$ aspect by a delta method. More precisely, let $\phi$ be a Hecke--Maass cusp form for $\rm SL(3,\mathbb{Z})$ and…

Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian \'etale covers of $X$ are…

We study minimal complex surfaces S of general type with q(S)=q and p_g(S)=2q-3, q>= 5. We give a complete classification in case that S has a fibration onto a curve of genus >=2. For these surfaces K^2=8\chi. In general we prove that…

We establish that every $K$-quasiconformal mapping of $w$ of the unit disk $\ID$ onto a $C^2$-Jordan domain $\Omega$ is Lipschitz provided that $\Delta w\in L^p(\ID)$ for some $p>2$. We also prove that if in this situation $K\to 1$ with…

We prove the so-called Severi inequality, stating that the invariants of a minimal smooth complex projective surface of maximal Albanese dimension satisfy: K^2_S >= 4\chi(S).

Let $S$ be a closed orientable hyperbolic surface with Euler characteristic$\chi$, and let $\lambda_k(S)$ be the $k$-th positive eigenvalue for the Laplacian on $S$. According to famous result of Otal and Rosas, $\lambda_{-\chi}>\frac14$.…

Given an irrational number $\alpha$ consider its irrationality measure function $\psi_{\alpha}(t)=\min\limits_{1\le q\le t, q\in\mathbb{Z}}\|q\alpha\|$. The set of all values of $\lambda(\alpha)=(\limsup\limits_{t\to\infty}…

Let $\pi$ be a Hecke cusp form for $\mathrm{SL}_3(\mathbb{Z})$. We bound the second moment average of $L(s,\pi)$ over a short interval to obtain the subconvexity estimate $$ L(1/2+it, \pi) \ll_{\pi, \varepsilon}…

Let $S$ be a compact hyperbolic surface of genus $g\geq 2$ and let $I(S) = \frac{1}{\mathrm{Vol}(S)}\int_{S} \frac{1}{\mathrm{Inj}(x)^2 \wedge 1} dx$, where $\mathrm{Inj}(x)$ is the injectivity radius at $x$. We prove that for any $k\in…