Osserman Conjecture in dimension n ? 8, 16

ct 2 00 2 Osserman Conjecture in dimension n 6 = 8 , 16

Let M n be a Riemannian manifold and R its curvature tensor. For a point p ∈ M n and a unit vector X ∈ TpM n , the Jacobi operator is defined by RX = R(X, ·)X. The manifold M n is called pointwise Osserman if, for every p ∈ M n , the spectrum of the Jacobi operator does not depend of the choice of X, and is called globally Osserman if it depends neither of X, nor of p. Osserman conjectured that...

Osserman manifolds of dimension 8

For a Riemannian manifold M n with the curvature tensor R, the Jacobi operator RX is defined by RX Y = R(X, Y)X. The manifold M n is called pointwise Osserman if, for every p ∈ M n , the eigenvalues of the Jacobi operator RX do not depend of a unit vector X ∈ TpM n , and is called globally Osserman if they do not depend of the point p either. R. Osserman conjectured that globally Osserman manif...

Osserman Conjecture In

On De Giorgi Conjecture in Dimension N ≥ 9

A celebrated conjecture due to De Giorgi states that any bounded solution of the equation ∆u+(1−u2)u = 0 in RN with ∂yN u > 0 must be such that its level sets {u = λ} are all hyperplanes, at least for dimension N ≤ 8. A counterexample for N ≥ 9 has long been believed to exist. Based on a minimal graph Γ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in RN , N ≥ 9, we prove t...

A short proof of the maximum conjecture in CR dimension one

In this paper and by means of the extant results in the Tanaka theory, we present a very short proof in the specific case of CR dimension one for Beloshapka's maximum conjecture. Accordingly, we prove that each totally nondegenerate model of CR dimension one and length >= 3 has rigidity. As a result, we observe that the group of CR automorphisms associated with each of such models contains onl...