Frederico Xavier

Efimov's theorem in dimension greater than two

Brian Smyth and Frederico Xavier Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA It is a classical theorem of Hilbert that the hyperbolic plane cannot be realized isometrically in ~3. In 1968 there appeared Efimovs celebrated generalization of this result [2] :

Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature

Abstract. Let

Real solvability of the equation ∂ 2/z ω = ρg and the topology of isolated umbilics

M^{2n}

RIGIDITY OF THE IDENTITY

be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if

Birationality of étale maps via surgery

M^{2n}

On the Structure of Complete Simply Connected Embedded Minimal Surfaces

is homeomorphic to a Kähler manifold, then its Euler number satisfies the inequality

INJECTIVITY OF LOCAL DIFFEOMORPHISMS FROM NEARLY SPECTRAL CONDITIONS

(-1)^n \chi(M^{2n})\geq 0

Convex hulls of complete minimal surfaces

.

Global inversion via the Palais-Smale condition

The geometric form of a conjecture associated with the names of Loewner and Carathéodory states that near an isolated umbilic in a smooth surface in ℝ3, the principal line fields must have index ≤ 1. Real solutions of the differential equation ∂ 2/z ω = g, where the complex function g is given only up to multiplication by a positive function, are intimately related to umbilics. We determine necessary and sufficient conditions of an integral nature for real solvability of this equation, which is really a system of two wave equations. We then construct germs of line fields of every index j ∈ 1/2 ℤ on S2 that cannot be realized as the Gauss image of the principal line fields near an isolated umbilic of positive curvature on any smooth surface in ℝ3. These include the standard dipole line field of index two and controlled distortions of it.

A Sharp Geometric Estimate for the Index of an Umbilic on a Smooth Surface

The structure of the group Aut(ℂn) of biholomorphisms of ℂn is largely unknown if n > 1. In stark contrast Aut(ℂ) is rather small, consisting of the non-constant affine linear maps. The description of Aut(ℂ) follows from the observation that an injective holomorphic function f : ℂ → ℂ satisfying f(0) = 0 and f′(0) = 1 must be the identity. These considerations suggest that similar characterizations of the identity might be useful in understanding the structure of Aut(ℂn). Using geometric methods we prove that an injective holomorphic map f : ℂn → ℂn is the identity I if and only if the power series at 0 of f - I has no terms of order ≤ 2n + 1 and the function |det Df(z)| |z|2n |f(z)|-2n is subharmonic throughout ℂn.