Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the "volume cone implies metric cone" theorem, the maximal diameter theorem, [Cg], and the splitting theorem, [CG]. Each asserts that if a certain geometric quantity (volume or diameter) is as large as possible relative to the pertinent lower bound on Ricci curvature, then the metric on the manifold in question is a warped product metric of a particular type. In this paper we provide quantitative generalizations of the above mentioned results. Among the applications are the splitting theorem for GromovHausdorff limit spaces X, where Mn -* X, Ricmn ? -i see [FY]. Other applications include the assertion that for complete manifolds, M', with Ricmf > 0 and Euclidean volume growth, all tangent cones at infinity are metric cones; compare [BKN], [CT], [P1]. Via resealing arguments, there are also strong consequences for the local structure of manifolds whose Ricci curvature satisfies a fixed lower bound and for their Gromov-Hausdorff limits. Some of these are announced in [CCol]; for a more detailed discussion see [CCo2], [CCo3], [CCo4]. Our work further develops and significantly extends techniques which were introduced in [Col], [Co2] and significantly extended in [Co3], in order to prove certain "stability" conjectures of Anderson-Cheeger, Gromov and Perelman. The results of [Col]-[Co3] were announced in [Co4]. We briefly review some of those results. Let dGH denote the Gromov-Hausdorff distance between metric spaces; see [GLP]. Let S' denote the unit sphere and recall that S' is the unique complete

Lower bounds on Ricci curvature and the almost rigidity of warped products

This paper, the second of a series, deals with the function space \mathcal{H} of all smooth Kahler metrics in any given n-dimensional, closed complex manifold V, these metrics being restricted to a given, fixed, real cohomology class, called a polarization of V. This function space is equipped with a pre-Hilbert metric structure introduced by T. Mabuchi [10], who also showed that, formally, this metric has nonpositive curvature. In the first paper of this series [4], the second author showed that the same space is a path length space. He also proved that \mathcal{H} is geodesically convex in the sense that, for any two points of \mathcal{H}, there is a unique geodesic path joining them, which is always length minimizing and of class C1,1. This partially verifies two conjectures of Donaldson [8] on the subject. In the present paper, we show first of all, that the space is, as expected, a path length space of nonpositive curvature in the sense of A. D. Aleksandrov. A second result is related to the theory of extremal Kahler metrics, namely that the gradient flow in \mathcal{H} of the "K energy" of V has the property that it strictly decreases the length of all paths in \mathcal{H}, except those induced by one parameter families of holomorphic automorphisms of M.

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The Space of Kähler Metrics

(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.

Another Report on Harmonic Maps

Entropies et rigidités des espaces localement symétriques de courbure strictement négative

We give examples of complete locally asymptotically flat Riemannian 4-manifolds with zero scalar curvature and negative mass. The generalized positive action conjecture of Hawking and Pope [5] is therefore false.

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Counter-examples to the generalized positive action conjecture

Let (M, J, g) be a compact Kahler manifold of constant scalar curvature. Then the Kahler class [ω] has an open neighborhood inH
1,1 (M, ℝ) consisting of classes which are represented by Kahler forms of extremal Kahler metrics; a class in this neighborhood is represented by the Kahler form of a metric of constant scalar curvature iff the Futaki invariant of the class vanishes. If, moreover, the derivative of the Futaki invariant at [ω] is “nondegenerate,” every small deformation of the complex manifold (M, J) also carries Kahler metrics of constant scalar curvature. We then apply these results to prove new existence theorems for extremal Kahler metrics on certain compact complex surfaces.

Extremal Kähler metrics and complex deformation theory

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Explicit self-dual metrics on $\mathbb{CP}_2 \# \cdots\#\mathbb{CP}_2$

Strong rigidity of positive quaternion Kähler manifolds.

It is shown that there are infinitely many compact simply con- nected smooth 4-manifolds which do not admit Einstein metrics, but nev- ertheless satisfy the strict Hitchin-Thorpe inequality 2 !> 3|" |. The exam- ples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems from Seiberg-Witten theory.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1996/0003/0002/MRL-1996-0003-0002-a001.pdf

Claude LeBrun

Papers

Counter-examples to the generalized positive action conjecture

Extremal Kähler metrics and complex deformation theory

Explicit self-dual metrics on $\mathbb{CP}_2 \# \cdots\#\mathbb{CP}_2$

Strong rigidity of positive quaternion Kähler manifolds.

Four-Manifolds without Einstein Metrics