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Methods Of Modern Mathematical Physics

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in R 3 is at least 2 2 . We prove this conjecture using the min-max theory of minimal surfaces.

Min-Max theory and the Willmore conjecture

On Seifert-manifolds

Lie groupoids and lie algebroids in differential geometry

We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.

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On the geometry of Riemannian manifolds with a Lie structure at infinity

We define and study an algebra Ψ ∞,0,V (M0) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞,0,V (M0). We also consider the algebra Diff ∗ (M0) of differential operators on M0 generated by V and C ∞ (M), and show that Ψ ∞,0,V (M0) is a microlocalization of Diff ∗ (M0). Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and “suspended” versions of the algebra Ψ ∞,0,V (M0).

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Pseudodifferential operators on manifolds with a Lie structure at infinity

Let us x a conformal class [g0] and a spin structure on a compact manifold M. For any g 2 [g0], let + (g) be the smallest positive eigenvalue of the Dirac operator D on (M; g; ). In a previous paper we have shown that + (M; g0; ) := inf g2[g0] + (g)vol(M; g) 1=n > 0: In the present article, we enlarge the conformal class by certain singular metrics. We will show that if + (M; g0; ) < + (S n ), then the inm um is attained on the enlarged conformal class. For proving this, we have to solve a system of semi-linear partial dieren tial equations involving a nonlinearity with critical exponent: D’ = j’j 2=(n 1) ’: The solution of this problem has many analogies to the solution of the Yamabe problem. However, our reasoning is more involved than in the Yamabe problem as the eigenvalues of the Dirac operator tend to +1 and 1 . Using the Weierstra representation, the solution of this equation in dimension 2 provides a tool for constructing new periodic constant mean curvature surfaces.

/pdf/the-smallest-dirac-eigenvalue-in-a-spin-conformal-class-and-2joj3oua36.pdf

The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions

We study some basic analytic questions related to dif- ferential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for Lie submanifolds allows us also to extend to regular open subsets of Lie manifolds the classical results on traces of functions in suitable Sobolev spaces. Our main application is a regularity result on poly- hedral domains P ⊂ R 3 using the weighted Sobolev spaces K m (P). In particular, we show that there is no loss of K m -regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify K m (P) with the Sobolev spaces on P associated to the metric r −2 P gE, where gE is the Euclidean metric and rP(x) is a smoothing of the Euclidean distance from x to the set of singular points of P. A suitable compactification of the interior of P then becomes a regular open subset of a Lie manifold. We also obtain the well-posedness of a non-standard boundary value problem on a smooth, bounded do- main with boundary O ⊂ R n using weighted Sobolev spaces, where the weight is the distance to the boundary.

/pdf/sobolev-spaces-on-lie-manifolds-and-regularity-for-3omxezsuvs.pdf

Sobolev spaces on lie manifolds and regularity for polyhedral domains

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to1 or there are eigenvalues converging to those of the torus. This is shown to be true in general for collapsing circle bundles with totally geodesic bers. Using the Hopf bration we use this fact to compute the Dirac eigenvalues on complex projective space including the multiplicities. Finally, we show that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have nonconstant spectrum. The marked length spectrum is also constant for these families.

/pdf/the-dirac-operator-on-nilmanifolds-and-collapsing-circle-3x834ffsbo.pdf

Bernd Ammann

Papers

On the geometry of Riemannian manifolds with a Lie structure at infinity

Pseudodifferential operators on manifolds with a Lie structure at infinity

The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions

Sobolev spaces on lie manifolds and regularity for polyhedral domains

The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles