Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix 0<α<1. Let N_α(d) denote the maximum number of lines through the origin in R^d with pairwise common angle arccos α. Let k denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly (1−α)/(2α). If k<∞, then N_α(d)= ⌊ k(d−1)/(k−1) ⌋ for all sufficiently large d, and otherwise N_α(d)=d+o(d). In particular, N_{1/(2k−1)}(d)= ⌊ k(d−1)/(k−1) ⌋ for every integer k≥2 and all sufficiently large d. A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.

We give a solution to the inverse problem (given a prescribed function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and Lp-compression functions of finitely generated groups. For smaller classes, we give solutions among solvable groups of exponential volume growth. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula relating the Lp-compression exponent of a group and its wreath product with the cyclic group forp in [1,2].

The purpose of this paper is to demonstrate in a rather general setup how isoperimetric inequalities and lower bounds of the eigenvalues of the Laplacian can be derived from existence of a distance function with controllable

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\"ahler) metric equation \cite{cc1} to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of K\"ahler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\mathcal{E}^1, d_1)$ are smooth. The continuity path proposed in \cite{chen15} is instrumental in this proof.

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

For a compact connected set$X$⊆$ℓ$^{∞}, we define a quantity$β$′($x$,$r$) that measures how close$X$may be approximated in a ball$B$($x$,$r$) by a geodesic curve. We then show that there is$c$>0 so that if$β$′($x$,$r$)>$β$>0 for all$x$∈$X$and$r$<$r$_{0}, then $\operatorname{dim}X>1+c\beta^{2}$ . This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.

Introduction. The classical Schwarz-Pick lemma states that any holomorphic map of the unit disk into itself decreases the Poincare metric. Later Ahlfors generalized this lemma to holomorphic mappings between two Riemann surfaces where curvatures of these Riemann surfaces were used in a very explicit way. More recently, Chern initiated the study of holomorphic mappings between higher-dimensional complex manifold by generalizing the Ahlfors lemma to these spaces. Then this lemma was further extended by Kobayashi, Griffiths, Wu and others. It plays a very important role in their theory. In this note, we shall prove the following generalization of the Schwarz lemma.

Let $K$ be a self-similar set satisfying the open set condition. Following Kaimanovich’s elegant idea, it has been proved that on the symbolic space $X$ of $K$ a natural augmented tree structure $E$ exists; it is hyperbolic, and the hyperbolic boundary $\partial_H X$ with the Gromov metric is H\"older equivalent to $K$. In this paper we consider certain reversible random walks with return ratio $0 < \lambda < 1$ on $(X,E)$. We show that the Martin boundary ${\mathcal M}$ can be identified with $\partial_H X$ and $K$. With this setup and a device of Silverstein, we obtain precise estimates of the Martin kernel and the Na\"im kernel in terms of the Gromov product. Moreover, the Naïm kernel turns out to be a jump kernel satisfying the estimate $\Theta (\xi,\eta) \asymp |\xi-\eta|^{-(\alpha+\beta)}$, where $\alpha$ is the Hausdorff dimension of $K$ and $\beta$ depends on $\lambda$. For suitable $\beta$, the kernel defines a regular non-local Dirichlet form on $K$. This extends the results of Kigami concerning random walks on certain trees with Cantor-type sets as boundaries.

We find a concise relation between the moduli $\tau , \rho$ of a rational Narain lattice $\Gamma (\tau, \rho)$ and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.

In this article, we continue the work in Guan-Li and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains.