Related papers: The Abresch-Rosenberg Shape Operator and applicati…

We propose a definition of differential operators of an associative algebra $A$ in the spirit of Hochschild cohomology. Specifically we define $D(A)$ as the zero cohomology of a certain bicomplex formed by Hom-spaces…

Let $\Gamma \subset \mathbb{R}^d$ be a smooth curve containing the origin. Does every Borel subset of $\mathbb R^d$ of sufficiently small codimension enjoy a S\'ark\"ozy-like property with respect to $\Gamma$, namely, contain two elements…

For every positive integer $N$ we determine the Enriques--Kodaira type of the Humbert surface of discriminant $N^2$ which parametrises principally polarised abelian surfaces that are $(N,N)$-isogenous to a product of elliptic curves. A key…

We obtain compact orientable embedded surfaces with constant mean curvature $0<H<\frac{1}{2}$ and arbitrary genus in $\mathbb{S}^2\times\mathbb{R}$. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean…

We consider a smooth hyper-surface Z of a closed Riemannian manifold X. Let P be the Poisson operator associating to a smooth function on Z its harmonic extension on X\Z. If A is a pseudo-differential operator on X of degree <3, we prove…

In $L_2 (\mathbb{R}^d; \mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$ with periodic coefficients depending on $\mathbf{x}/\varepsilon$. We find approximations…

In this article we consider solvable hypersurfaces of the form $N \exp(\R H)$ with induced metrics in the symmetric space $M = SL(3,\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition…

We introduce the notion of the \textit{Bruce-Roberts number} for holomorphic 1-forms relative to complex analytic varieties. Our main result shows that the Bruce-Roberts number of a 1-form $\omega$ with respect to a complex analytic…

We study a surface discovered by Stover which is the surface with minimal Euler number and maximal automorphism group among smooth arithmetic ball quotient surfaces. We study the natural map $\wedge^{2}H^{1}(S,\mathbb{C})\to…

We consider an entire graph $S$ in $\mathbb R^{N+1}$ of a continuous real function $f$ over $\mathbb R^{N}$ with $N\ge 1$. Let $\Omega$ be an unbounded domain in $\mathbb R^{N+1}$ with boundary $S$. Consider nonlinear diffusion equations of…

We study the critical set C of the nonlinear differential operator F(u) = -u" + f(u) defined on a Sobolev space of periodic functions H^p(S^1), p >= 1. Let R^2_{xy} \subset R^3 be the plane z = 0 and, for n > 0, let cone_n be the cone x^2 +…

The $H$-space, denoted as $(\mathbb{R}, \tau_{A})$, has $\mathbb{R}$ as its point set and a basis consisting of usual open interval neighborhood at points of $A$ while taking Sorgenfrey neighborhoods at points of $\mathbb{R}$-$A$. In this…

We study singular integral operators induced by $3$-dimensional Calder\'on-Zygmund kernels in the Heisenberg group. We show that if such an operator is $L^{2}$ bounded on vertical planes, with uniform constants, then it is also $L^{2}$…

In [dLMu05], DeLellis and M\"uller proved a quantitative version of Codazzi's theorem, namely for a smooth embedded surface $\ \Sigma \subseteq \mathbb{R}^3\ $ with area normalized to $\ {\cal H}^2(\Sigma) = 4 \pi\ $, it was shown that $\…

We construct a new minimal complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (in fact $\pi_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existence question for numerical Campedelli surfaces…

Let $ X = \Gamma\setminus \mathbb{H} $ be a non-elementary geometrically finite hyperbolic surface and let $ \delta $ denote the Hausdorff dimension of the limit set $ \Lambda(\Gamma) $. We prove that for every $ \varepsilon > 0 $ the…

We first construct a real family of $SL(2,\mathbb{R})$-invariant symbol composition product $\{\sharp_\theta\}_{\theta\in,\mathbb{R}}$ on the analogue of the Schwartz space $S(\mathbb{D})$ on the hyperbolic plane…

Let $X$ be a compact Riemann surface of genus $g\geq 2$ equipped with flat conical metric $|\Omega|$, where $\Omega$ be a holomorphic quadratic differential on $X$ with $4g-4$ simple zeroes. Let $K$ be the canonical line bundle on $X$.…

We consider self-adjoint Schr\"odinger operators in $L^2 (\mathbb{R}^d)$ with a $\delta$-interaction of strength $\alpha$ and a $\delta'$-interaction of strength $\beta$, respectively, supported on a hypersurface, where $\alpha$ and…

We study the boundedness problem for maximal operators in 3-dimensional Euclidean space associated to hypersurfaces given as the graph of $c+f$, where $f$ is a mixed homogeneous function which is smooth away from the origin and $c$ is a…