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In this paper, we will study the (linear) geometric analysis on metric measure spaces. We will establish a local Li-Yau’s estimate for weak solutions of the heat equation and prove a sharp Yau’s gradient gradient for harmonic functions on metric measure spaces, under the Riemannian curvature-dimension condition RCD(K; N).
In 1997, J. Jost  and F. H. Lin , independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally H¨older continuous. In , F. H. Lin proposed a challenge problem: Can the H¨older continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in ). The main theorem of this paper gives a complete resolution to it.
In a symmetric space of noncompact type X = G=K oriented geodesic segments correspond modulo isometries to vectors in the
Euclidean Weyl chamber. We can hence assign vector valued lengths to segments. Our main result is a system of homogeneous linear inequalities, which we call the generalized triangle inequalities or stability inequalities, describing the restrictions on
the vector valued side lengths of oriented polygons. It is based on the mod 2 Schubert calculus in the real Grassmannians G=P for
maximal parabolic subgroups P.
The side lengths of polygons in Euclidean buildings are studied in the related paper [KLM2]. Applications of the geomet-
ric results in both papers to algebraic group theory are given in[KLM3].
In the ¯rst Heisenberg group H1 with its sub-Riemannian structure generated by the horizontal subbundle, we single out a class
of C2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have
vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C2 entire Hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in H1 the only stable C2 Hminimal entire graphs, with empty characteristic locus, are thevertical planes.