Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.     

Toric partial density functions and stability of toric varieties

Let $(L, h)\rightarrow (X, \omega )$ denote a polarized toric Kähler manifold. Fix a toric submanifold $Y$ and denote by $\hat{\rho }_{tk}:X\rightarrow \mathbb {R}$ the partial density function corresponding to the partial Bergman kernel projecting smooth sections of $L^k$ onto holomorphic sections of $L^k$ that vanish to order at least $tk$ along $Y$ , for fixed $t>0$ such that $tk\in \mathbb {N}$ . We prove the existence of a distributional expansion of $\hat{\rho }_{tk}$ as $k\rightarrow \infty $ , including the identification of the coefficient of $k^{n-1}$ as a distribution on $X$ . This expansion is used to give a direct proof that if $\omega $ has constant scalar curvature, then $(X, L)$ must be slope semi-stable with respect to $Y$ (cf. Ross and Thomas in J Differ Geom 72(3): 429–466, 2006). Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.     

A Categorical Approach to Groupoid Frobenius Algebras

In this paper, we show that $\mathcal{G}$ -Frobenius algebras (for $\mathcal{G}$ a finite groupoid) correspond to a particular class of Frobenius objects in the representation category of $D(k[\mathcal{G}])$ , where $D(k[\mathcal{G}])$ is the Drinfeld double of the quantum groupoid $k[\mathcal{G}]$ (Nikshych et al. 2000).     

The Diederich–Fornaess index and the global regularity of the \bar{\partial }-Neumann problem

We describe along the guidelines of Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), the constant ${\mathcal {E}}_s$ which is needed to control the commutator of a totally real vector field $T_{{\mathcal {E}}}$ with $\bar{\partial }^*$ in order to have $H^s$ a-priori estimates for the Bergman projection $B_k, k\ge q-1$ , on a smooth $q$ -convex domain $D\subset \subset {\mathbb {C}}^{n}$ . This statement, not explicit in Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), yields regularity of $B_k$ in specific Sobolev degree $s$ . Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function $r$ of $D$ , the operators $(T^+)^{-\frac{\delta }{2}}$ and $(-r)^{\frac{\delta }{2}}$ . We are thus able to extend to general degree $k\ge 0$ of $B_k$ , the conclusion of (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999) which only holds for $q=1$ and $k=0$ : if for the Diederich–Fornaess index $\delta $ of $D$ , we have $(1-\delta )^{\frac{1}{2}}\le {\mathcal {E}}_s$ , then $B_k$ is $H^s$ -regular.     

Characterization of Siegel cusp forms by the growth of their Fourier coefficients

We characterize all Siegel cusp forms of degree $n$ and large weight $k$ by the growth of their Fourier coefficients. More precisely we prove, among other related results, that if the Fourier coefficients of a modular form on the congruence subgroup $\Gamma _0^n(N)$ of square–free level $N$ satisfy the “Hecke bound” at the cusp $\infty $ , then it must be a cusp form, provided $k >2n+1$ .     

Fundamental groups of positively curved manifolds with symmetry

We obtain a classification for the fundamental groups of closed $n$ -manifolds of positive sectional curvature which admit an isometric $T^k$ -action with $k \ge \frac{n}{6}+1 (n \ne 11, 15)$ .     

On semidefinite programming relaxations of maximum k-section

We derive a new semidefinite programming bound for the maximum $k$ -section problem. For $k=2$ (i.e. for maximum bisection), the new bound is at least as strong as a well-known bound by Poljak and Rendl (SIAM J Optim 5(3):467?C487, 1995). For $k \ge 3$ the new bound dominates a bound of Karisch and Rendl (Topics in semidefinite and interior-point methods, 1998). The new bound is derived from a recent semidefinite programming bound by De Klerk and Sotirov for the more general quadratic assignment problem, but only requires the solution of a much smaller semidefinite program.     

Explicit Constructions of Centrally Symmetric k-Neighborly Polytopes and Large Strictly Antipodal Sets

We present explicit constructions of centrally symmetric $2$ -neighborly $d$ -dimensional polytopes with about $3^{d/2}\approx (1.73)^d$ vertices and of centrally symmetric $k$ -neighborly $d$ -polytopes with about $2^{{3d}/{20k^2 2^k}}$ vertices. Using this result, we construct for a fixed $k\ge 2$ and arbitrarily large $d$ and $N$ , a centrally symmetric $d$ -polytope with $N$ vertices that has at least $\left( 1-k^2\cdot (\gamma _k)^d\right) \genfrac(){0.0pt}{}{N}{k}$ faces of dimension $k-1$ , where $\gamma _2=1/\sqrt{3}\approx 0.58$ and $\gamma _k = 2^{-3/{20k^2 2^k}}$ for $k\ge 3$ . Another application is a construction of a set of $3^{\lfloor d/2 -1\rfloor }-1$ points in $\mathbb R ^d$ every two of which are strictly antipodal as well as a construction of an $n$ -point set (for an arbitrarily large $n$ ) in $\mathbb R ^d$ with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.     

The Laplacian on Cylindrical Domains

The aim of this paper is to prove elliptic regularity and parabolic maximal regularity of the Laplacian with mixed boundary conditions on domains Ω carrying a cylindrical structure. More precisely, we consider Ω to be given as the Cartesian product of whole or half spaces, a cube ${\mathcal{Q}}$ , and a standard domain V having compact boundary. Taking advantage of this structure we apply operator-valued Fourier multiplier results to transfer ${\mathcal{H}^{\infty}}$ -calculus results known for the Laplacian in L ^p (V) to the Laplacian in L ^p (Ω). This approach turns out to inherit elliptic regularity, i.e. the domain of the Dirichlet Laplacian equals ${W^{2,p}(\Omega) \cap W_0^{1,p}(\Omega)}$ , for instance. This is surprising since Ω may be unbounded and non-convex with boundary neither compact nor of class C ^1,1 at the same time. More generally, we consider the following mixture of boundary conditions: on every smooth part of the boundary Dirichlet or Neumann boundary conditions are imposed and on parts related to ${\mathcal{Q}}$ generalized periodic boundary conditions are included. Via ${\mathcal{R}}$ -sectoriality we deduce maximal regularity in the parabolic sense which seems to be new for this general class of boundary conditions. Parabolic equations with such a mixture of boundary conditions on such type of domains appear for example in models describing growth of biological cells.     

Generalized Strong Boundary Points and Boundaries of Families of Continuous Functions

If ${\mathcal{A}}$ is a family of continuous functions on a locally compact Hausdorff space X, a boundary for ${\mathcal{A}}$ is a subset ${B \subset X}$ such that every ${f \in \mathcal{A}}$ attains its maximum modulus on B. In this work we generalize the concept of strong boundary points for families of functions and show that the collection of these generalized strong boundary points is always a boundary for ${\mathcal{A}}$ . We give conditions under which all boundaries for ${\mathcal{A}}$ consist of generalized strong boundary points and under which these points coincide with classical strong boundary points. When ${\mathcal{A}}$ has sufficient algebraic structure it is proven that this construction provides a unique boundary for ${\mathcal{A}}$ consisting of boundary points, and we conclude by demonstrating how this approach provides an alternate technique for proving the existence of the Choquet and Shilov boundaries in certain function algebras.     

Uniqueness for Inverse Boundary Value Problems by Dirichlet-to-Neumann Map on Subboundaries

We consider inverse boundary value problems for elliptic equations of second order of determining coefficients by Dirichlet-to-Neumann map on subboundaries, that is, the mapping from Dirichlet data supported on subboundary ${\partial \Omega \setminus \Gamma_{-}}$ to Neumann data on subboundary ${\partial \Omega \setminus \Gamma_{+}}$ . First we prove uniqueness results in three dimensions under some conditions such as ${\overline{\Gamma_{+}\cup\Gamma_{-}}= \partial\Omega}$ Next we survey uniqueness results in two dimensions for various elliptic systems for arbitrarily given ${\Gamma_{-} = \Gamma_{+}}$ Our proof is based on complex geometric optics solutions which are constructed by a Carleman estimate.     

Quasilinear elliptic and parabolic Robin problems on Lipschitz domains

We prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the p-Laplace operator for all ${p \in (1,\infty)}$ , but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is well-posed in the space ${\mathrm{C}(\overline{\Omega})}$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in ${\mathrm{C}(\overline{\Omega})}$ is m-accretive and densely defined. Thus it generates a non-linear strongly continuous contraction semigroup on ${\mathrm{C}(\overline{\Omega})}$ .     

The Saito–Kurokawa lifting and Darmon points

Let $E_{/_\mathbb{Q }}$ be an elliptic curve of conductor $Np$ with $p\not \mid N$ and let $f$ be its associated newform of weight $2$ . Denote by $f_\infty $ the $p$ -adic Hida family passing though $f$ , and by $F_\infty $ its $\varLambda $ -adic Saito–Kurokawa lift. The $p$ -adic family $F_\infty $ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients $\{\widetilde{A}_T(k)\}_T$ indexed by positive definite symmetric half-integral matrices $T$ of size $2\times 2$ . We relate explicitly certain global points on $E$ (coming from the theory of Darmon points) with the values of these Fourier coefficients and of their $p$ -adic derivatives, evaluated at weight $k=2$ .     

Generalized normal homogeneous Riemannian metrics on spheres and projective spaces

In this paper, we develop new methods to study generalized normal homogeneous Riemannian manifolds. In particular, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on spheres ${S^n}$ . We prove that for any connected (almost effective) transitive on $S^n$ compact Lie group $G$ , the family of $G$ -invariant Riemannian metrics on $S^n$ contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and $n\ge 5$ . Any such family (that exists only for $n=2k+1$ ) contains a metric $g_\mathrm{can}$ of constant sectional curvature $1$ on $S^n$ . We also prove that $(S^{2k+1}, g_\mathrm{can})$ is Clifford–Wolf homogeneous, and therefore generalized normal homogeneous, with respect to $G$ (except the groups $G={ SU}(k+1)$ with odd $k+1$ ). The space of unit Killing vector fields on $(S^{2k+1}, g_\mathrm{can})$ from Lie algebra $\mathfrak g $ of Lie group $G$ is described as some symmetric space (except the case $G=U(k+1)$ when one obtains the union of all complex Grassmannians in $\mathbb{C }^{k+1}$ ).     

The {\bar{\partial}} -Neumann problem on product domains in {\mathbb{C}^{n}}

Let ${\Omega=\Omega_{1}\times\cdots\times\Omega_{n}\subset\mathbb{C}^{n}}$ , where ${\Omega_{j}\subset\mathbb{C}}$ is a bounded domain with smooth boundary. We study the solution operator to the ${\overline\partial}$ -Neumann problem for (0,1)-forms on Ω. In particular, we construct singular functions which describe the singular behavior of the solution. As a corollary our results carry over to the ${\overline\partial}$ -Neumann problem for (0,q)-forms. Despite the singularities, we show that the canonical solution to the ${\overline\partial}$ -equation, obtained from the Neumann operator, does not exhibit singularities when given smooth data.     

Bounds for the blowup time of the solutions to quasi-linear parabolic problems

In this paper, we obtain the lower and the upper bounds of the blowup time of the solutions to quasi-linear parabolic problems subject to Dirichlet(or Neumann) boundary condition. Our results are suitable for the problems with any smooth bounded domain ${\Omega \subset \mathbb{R}^n}$ and ${n \geq 3}$ . In some special cases, we can even get the exact values of blowup time.     

Diagram Groupoids and von Neumann Algebras

The main purpose of this paper is to study certain algebraic structures induced by directed graphs. We have studied graph groupoids, which are algebraic structures induced by given graphs. By defining a certain groupoid-homomorphism ?? on the graph groupoid ${\mathbb{G}}$ of a given graph G, we define the diagram of G by the image ${\delta(\mathbb{G})}$ of ??, equipped with the inherited binary operation on ${\mathbb{G}}$ . We study the fundamental properties of the diagram ${\delta(\mathbb{G})}$ , and compare them with those of ${\mathbb{G}}$ . Similar to Cho (Acta Appl Math 95:95?C134, 2007), we construct the groupoid von Neumann algebra ${\mathcal{M}_{G}=vN(\delta(\mathbb{G}))}$ , generated by ${\delta(\mathbb{G})}$ , and consider the operator algebraic properties of ${\mathcal{M}_{G}}$ . In particular, we show ${\mathcal{M}_{G}}$ is *-isomorphic to a von Neumann algebra generated by a family of idempotent operators and nilpotent operators, under suitable representations.