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We consider manifolds with conic singularities that are isometric to Rn outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author [23] to establish a “very weak” Huygens’ principle, which may be of independent interest. As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schr¨odinger equation.

This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott’s original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperk¨ahler Kirwan map is surjective for the non-fixed determinant case.

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The Cauchy problem for the homogeneous real/complexMongeAmp`ere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of K¨ahler metrics equipped with the Mabuchi metric. This Cauchy problem is believed to be ill-posed and a basic problem is to characterize initial data of (weak) solutions which exist up to time T . In this article, we use a quantization method to construct a subsolution of the HCMA on a general projective variety, and we conjecture that it solves the equation for as long as the unique solution exists. The subsolution, called the “quantum analytic continuation potential,” is obtained by (i) Toeplitz quantizing the Hamiltonian flow determined by the Cauchy data, (ii) analytically continuing the quantization, and (iii) taking a certain logarithmic classical limit. We then prove that in the case of torus invariant metrics (where the HCMA reduces to the HRMA) the quantum analytic continuation potential coincides with the well-known Legendre transform potential, and hence solves the equation as long as it is smooth. In the sequel [29], it is proved that the Legendre transform potential ceases to solve the HCMA once it ceases to be smooth. The results here and in the sequels in particular characterize the initial data of smooth geodesic rays.