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In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evolves almost in the same way as that of scalar curvature along the surface Ricci flow obtained by Hamilton \cite{Ham1}. Then we study the long time behavior of the combinatorial Ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive Euler number. We further generalize the combinatorial curvature to $\alpha$-curvature and prove that it is also globally rigid, which is in fact a generalized Bower-Stephenson conjecture \cite{BS}. We also use the combinatorial Ricci flow to study the corresponding $\alpha$-Yamabe problem.

We prove families of uniform ( L r , L s ) resolvent estimates for simply connected manifolds of constant curvature (negative or positive) that imply the earlier ones for Euclidean space of Kenig, Ruiz and the second author [7] . In the case of the sphere we take advantage of the fact that the half-wave group of the natural shifted Laplacian is periodic. In the case of hyperbolic space, the key ingredient is a natural variant of the Stein–Tomas restriction theorem.

This paper develops necessary and sufficient conditions for pointwise inversion of Fourier transforms on rank one symmetric spaces of non-compact type for functions in the piecewise smooth category. This extends results of Pinsky for isotropic Riemannian manifolds of constant curvature. Methodologically, a similar result on the Jacobi transform and a transplantation scheme from even dimensional spaces to certain odd dimensional real hyperbolic spaces naturally enter the picture.

For any two-dimensional Riemannian manifold ( M , g ) we introduce a new functional, h g , on the space of closed simple nonparametrized curves on M . This functional associates to any simple curve Γ the regularize determinant of the Laplace operator on the manifold obtained by cutting M along Γ and imposing Dirichlet boundary conditions. When M is of genus zero we derive a formula for the variation of h g , we prove that the critical points are conformal circles (i.e., the curves which, with respect to the unique metric of constant curvature 1 in the conformal class { e 2α g :α ∈ C ∞ ( S 2 , R )} of g , have constant geodesic curvature), and that the hessian of the functional at a critical point is nondegenerate in directions normal the critical submanifold (Theorem 1.1). We also construct smooth flows on the space of nonparametrized curves retracting the space onto the critical sub-manifold and show that they are gradient-like for our function. These flows deform a given closed embedded curve on S 2 to a conformal circle keeping the area of the domain bounded by each curve of the deformation constant (Theorem 1.3).

We show in the context of integral currents that Poincare's isoperimetric variational problem for simple closed geodesics on ovaloids has a smooth extremal C without self-intersection. Provided the smooth Riemannian metric on the ovaloid M in question does not deviate too far from constant curvature, we further show that (i) this extremal C is connected and so is the desired non-trivial geodesic of shortest length on M and (ii) C is close (in the sense of Hausdorff distance) to a great circle.

By writing the Fourier inversion formula on Euclidean space in polar coordinates, we obtain f(x) = ∝0∞ fgl(x) dx, where Δfgl(x) = −λ2fgl(x), which is the spectral theory of the Laplacian. How do properties of f relate to properties of the family of eigenfunctions fgl? Answers are provided for the following spaces: L2, S, S′, D (in odd dimensions). Analogous results are obtained for harmonic analysis on hyperbolic space, constant curvature semi-Riemannian spaces, the Heisenberg group, and for differential forms on hyperbolic space. For f ϵ L2 there is a “Plancherel Formula” ‖f‖22=πlimt → ∞∫o∞1t∫bt(z) |fκ(x)|2dx dκ where Bt(z) denotes the ball of radius t about an arbitrary point z, which is independent of the dimension, and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted. For certain semi-Riemannian symmetric spaces, we obtain a “Paley-Wiener Theorem” that explains the role of discrete series in producing functions of compact support, and also involve certain non-unitary representations that contain the discrete series representations. For the Heisenberg group the “Plancherel Formula” is of a different nature, requiring that the eigenfunctions be almost periodic in one variable.

A general theory of quantization has been proposed by Berezin (see, e.g., his survey in Comm. Math. Phys. 40 (1975), 153-174). In this paper we establish a weak form of the correspondence principle for the annulus, quantized according to Berezin. More precisely, we show that B ħ → I as ħ → 0, where I is the identity operator and B ħ the Berezin transform. We consider also spectral analysis on the annulus. In particular, we express the eigenfunctions of the relevant Laplacian in terms of certain Romanovski polynomials. Finally, we write down the expression for the analogue of the Morse potential in this case. We remark that similar considerations can be made, in principle, on any circular domain in the presence of a radial Hermitian metric of constant curvature.

An explicit solution is given to the Cauchy problem for the source-free Maxwell's equations in a vacuum on a space-time of the form R 1 X M3, where M3 is a 3-manifold of constant curvature. This solution satisfies Huyghens' Principle, that all electromagnetic radiation propagates at exactly the speed of light. The solution is obtained by harmonic analysis on M3, and in the process a generating class of plane wave solutions is found. These solutions approximate the flat-space plane wave solutions in a neighbourhood of a point, but their global properties are somewhat different. The solutions obtained are easily transplanted to the Robertson-Walker models of General Relativity by re-scaling the time variable.

Let Rn2 denote the n-fold covering space of the plane with the origin deleted (0 < n ⩽ ∞), parametrized by polar coordinates (r, θ) with θ periodic of period 2πn, and equipped with the flat Riemannian metric dr2 + r2 dθ2. There is a natural harmonic analysis, with the Fourier transform given by f(c, λ) = 12π∫− ∞∞∫0∞ f(r, θ) J|v|(λr) eivθr dr dθ , Fourier inversion formula f(c, θ) = ∫− ∞∞∫0∞ f(r, θ) J|v|(λr) eivθλ dλ dv , and Plancherel formula ‖f‖22=2π ∫− ∞∞∫0∞ |f(v, λ)|2 λ dλ dv in the case n = ∞, and analogous expressions with v restricted to the values kn when n is finite. This harmonic analysis realizes the joint spectral decompositions of the self-adjoint operators i(∂∂θ) and Δ, where Δ is the Dirichlet or Friedrichs extension of the Laplace-Beltrami operator defined on D. This paper gives a detailed development of this harmonic analysis, including the effect of basic differentiation and multiplication operators on the Fourier transform, the theory of L2 Sobolev spaces Hk, and the distinctions between H2, the domain of Δ, and the space of functions f with both f and Δf in L2. In the process, it is necessary to study certain generalizations of the Hilbert transform that are related to the theory of Hankel transforms. Two applications of this harmonic analysis are given, to the study of a natural Radon transform (or X-ray transform) on Rn2, and to the solution of the heat equation on Rn2. The latter also gives information about the heat equation in a sector of the plane with Dirichlet or Neumann boundary conditions, and about Brownian motion in the plane taking into account the winding number of a Brownian path with respect to the origin. Analogous problems for spaces of non-zero constant curvature are discussed, and in this context some higher dimensional spaces arise naturally.