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A salami is a connected, locally finite, weighted graph with non-negative Ollivier Ricci curvature and at least two ends of infinite volume. We show that every salami has exactly two ends and no vertices with positive curvature. We moreover show that every salami is recurrent and admits harmonic functions with constant gradient. The proofs are based on extremal Lipschitz extensions, a variational principle and the study of harmonic functions. Assuming a lower bound on the edge weight, we prove that salamis are quasi-isometric to the line, that the space of all harmonic functions has finite dimension, and that the space of subexponentially growing harmonic functions is two-dimensional. Moreover, we give a Cheng-Yau gradient estimate for harmonic functions on balls.
We prove that the geodesic equation for any semi-Riemannian metric of regularity $C^{0,1}$ possesses $C^1$-solutions in the sense of Filippov.
Let K be a knot in the 3-sphere. A slope p/q is said to be characterising for K if whenever p/q surgery on K is homeomorphic, via an orientation-preserving homeomorphism, to p/q surgery on another knot K in the 3-sphere, then K and K are isotopic. It
We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we show that the
We prove that every tetrahedron T has a simple, closed quasigeodesic that passes through three vertices of T. Equivalently, every T has a face whose exterior angles are at most pi.
A graph $G$ with a group $H$ of automorphisms acting semiregularly on the vertices with two orbits is called a {em bi-Cayley graph} over $H$. When $H$ is a normal subgroup of $Aut(G)$, we say that $G$ is {em normal} with respect to $H$. In this paper