Consider the heat kernel $\wp (t,x,y)$ on the universal cover $\tilde{M}$ of a closed Riemannian manifold of negative sectional curvature. We show the local limit theorem for ℘:

$$\underset{t\to \mathrm{\infty }}{lim}{t}^{3∕2}{e}^{{\mathit{\lambda }}_{0}t}\wp (t,x,y)=C(x,y),$$

where ${\mathit{\lambda }}_0$ is the bottom of the spectrum of the geometric Laplacian and $C(x,y)$ is a positive ${\mathit{\lambda }}_0$-harmonic function which depends on $x,y\in \tilde{M}$. We also show that the ${\mathit{\lambda }}_0$-Martin boundary of $\tilde{M}$ is equal to its topological boundary. The Martin decomposition of $C(x,y)$ gives a family of measures $\{{\mathrm{\mu }}_x^{{\mathit{\lambda }}_0}\}$ on $\partial \tilde{M}$. We show that $\{{\mathrm{\mu }}_x^{{\mathit{\lambda }}_0}\}$ is a family minimizing the energy or Mohsen’s Rayleigh quotient. We apply the uniform Harnack inequality on the boundary $\partial \tilde{M}$ and the uniform three-mixing of the geodesic flow on the unit tangent bundle $SM$ for suitable Gibbs–Margulis measures.

We find a compactification of the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of projectivized geodesic currents of the space of flat metrics induced by holomorphic cubic differentials on a Riemann surface.

We answer a question of Friedlander, Iwaniec, Mazur, and Rubin on the joint distribution of spin symbols. As an application we give a negative answer to a conjecture of Cohn and Lagarias on the existence of governing fields for the 16-rank of class groups under the assumption of a short character sum conjecture.

The aim of this article is to give for each dimension $d\ge 2$ an infinite series of rigid compact complex manifolds which are not infinitesimally rigid and, hence, to give an exhaustive answer to a problem of Morrow and Kodaira stated in the famous book Complex Manifolds.

Our goal here is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive Lie groups. We first prove some general results on the existence of equivariant deformation quantizations of vector bundles on closed Lagrangian subvarieties, which lie in smooth symplectic varieties with Hamiltonian group actions. Then we apply them to the orbit method and construct nontrivial irreducible Harish-Chandra modules for certain nilpotent coadjoint orbits. Our examples include new geometric construction of representations associated to a large class of nilpotent orbits of real exceptional Lie groups.

Let X be a smooth curve over a finitely generated field k, and let ℓ be a prime different from the characteristic of k. We analyze the dynamics of the Galois action on the deformation rings of mod ℓ representations of the geometric fundamental group of X. Using this analysis, we prove several finiteness results for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey–Mazur conjecture for function fields in characteristic 0. For example, we show that if X is a normal, connected variety over $\mathbb{C}$, then the (typically infinite) set of representations of ${\mathrm{\pi }}_1(X^{\mathrm{an}})$ into ${\mathrm{GL}}_n(\overline{{\mathbb{Q}}_{\ell }})$, which come from geometry, has no limit points. As a corollary, we deduce that if L is a finite extension of ${\mathbb{Q}}_{\ell }$, then the set of representations of ${\mathrm{\pi }}_1(X^{\mathrm{an}})$ into ${\mathrm{GL}}_n(L)$, which arise from geometry, is finite.