LIPSCHITZ EXTENSION OF MULTIPLE BANACH-VALUED FUNCTIONS IN THE SENSE OF ALMGREN JORDAN GOBLET

Abstract. For any positive integer Q, a Q Q (Y)-valued function f on X is essentially a rule assigning Q unordered and not necessarily distinct elements of Y to each element of X. Equivalently f maps X into the space QQ(Y) of Q unordered points in Y. We study the Lipschitz extension problem in this context by using two general Lipschitz extension theorems recently proved by U. Lang and T. Schlichenmaier. We prove that the pair ( X, QQ(Y)) has the Lipschitz extension property if Y is a Banach space and X is a metric space with a finite Nagata dimension. We also show that QQ(Y) is an absolute Lipschitz retract if Y is a finite algebraic dimensional Banach space. 1

INVARIANT MEASURES UNDER GEODESIC FLOW HAMID-REZA FANA Ï

Abstract. For a compact Riemannian manifold with negative curvature, the Liouville measure, the Bowen-Margulis measure and the Harmonic measure are three natural invariant measures under the geodesic flow. We show that if any two of the above three measure classes coincide then the space is locally symmetric, provided the function with respect to which the equilibrium state is the Harmonic measure, depends only on the foot points. 1. Preliminaries Let (M, g) be a compact Riemannian manifold with strictly negative curvature. Its geodesic flow is of Anosov type and there are invariant probability measures under the geodesic flow on SM, the unit tangent bundle of M. Among them, three measures are extremely well-known: the Liouville measure, the Bowen-Margulis measure and the Harmonic measure. Let us describe them briefly. The Liouville measure m is the natural Liouville invariant measure corresponding to the contact structure of the geodesic flow, f dm = f(v) dv dx, SM M where dx is the (normalized) Riemannian volume on M and dv is the (normalized) Lebesgue measure on SxM. It is well-known ([7]) that m is the equilibrium state of the Hölder function tr U(v), where U(v) is the second fundamental form of th

EMBEDDING COMPACT RIEMANN SURFACES IN 4-DIMENSIONAL RIEMANNIAN MANIFOLDS

Abstract. Any compact Riemann surface has a conformal model in any orientable Riemannian manifold of dimension 4. Precisely, we will prove that, given any compact Riemann surface, there is a conformally equivalent model in a prespecified orientable 4-dimensional Riemannian manifold. This result along with [10] now shows that a compact Riemann surface admits conformal models in any Riemannian manifold of dimension ≥ 3. 1