ar X iv : 1 40 6 . 73 09 v 1 [ m at h . M G ] 2 7 Ju n 20 14 On Klein ’ s So - called Non - Euclidean geometry

ar X iv : m at h / 06 09 23 6 v 1 [ m at h . M G ] 8 S ep 2 00 6 WEAK METRICS ON EUCLIDEAN DOMAINS

A weak metric on a set is a function that satisfies the axioms of a metric except the symmetry and the separation axioms. In the present paper we introduced a weak metric, called the Apollonian weak metric, on any subset of a Euclidean space which is either bounded or whose boundary is unbounded. We then relate this weak metric to some familiar metrics such as the Poincaré metric, the Klein-Hil...

ar X iv : m at h / 03 09 40 8 v 2 [ m at h . D G ] 1 2 Ju l 2 00 4 EINSTEIN METRICS ON SPHERES

ar X iv : 0 80 9 . 15 36 v 2 [ m at h . D G ] 1 5 Ju n 20 09 Tight Lagrangian surfaces in S 2 × S 2 ∗ Hiroshi

We determine all tight Lagrangian surfaces in S2×S2. In particular, globally tight Lagrangian surfaces in S2 × S2 are nothing but real forms.

ar X iv : 0 90 6 . 32 38 v 1 [ m at h . N T ] 1 7 Ju n 20 09 GEOMETRIC AND p - ADIC MODULAR FORMS OF HALF - INTEGRAL WEIGHT by Nick

ar X iv : 0 90 6 . 33 84 v 1 [ m at h . A P ] 1 8 Ju n 20 09 ENERGY DISPERSED LARGE DATA WAVE MAPS IN 2 + 1 DIMENSIONS

In this article we consider large data Wave-Maps from R into a compact Riemannian manifold (M, g), and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent. This is a companion to our concurrent article [21], which together with the present work establishes a full regularity theory for large data Wave-Maps.