Cavitational flows and global injectivity of conformal maps

Continuity and Injectivity of Optimal Maps

Figalli–Kim–McCann proved in [14] the continuity and injectivity of optimal maps under the assumption (B3) of nonnegative cross-curvature. In the recent [15, 16], they extend their results to the assumption (A3w) of Trudinger-Wang [34], and they prove, moreover, the Hölder continuity of these maps. We give here an alternative and independent proof of the extension to (A3w) of the continuity and...

Approximate Maps Characterizing Injectivity and Surjectivity of Maps

In the theory of inverse systems, in order to study the properties of a space X or a map f : X → Y between spaces, one expands X to an inverse system X or expands f to a map f : X → Y between the inverse systems, and then work on X or f . In this paper, we define approximate injectivity (resp., surjectivity) for approximate maps, and show that a map f : X → Y between compact metric spaces is in...

Hölder continuity and injectivity of optimal maps∗

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x, y). If the source density f(x) is bounded away from zero and infinity in an open region U ′ ⊂ R, and the target density f−(y) is bounded away from zero and infinity on its support V ⊂ R, which is strongl...

Global Division of Cohomology Classes via Injectivity

The aim of this note is to remark that the injectivity theorems of Kollár and EsnaultViehweg can be used to give a quick algebraic proof of a strengthening (by dropping the positivity hypothesis) of the Skoda-type division theorem for global sections of adjoint line bundles vanishing along suitable multiplier ideal sheaves proved in [EL], and to extend this result to higher cohomology classes a...

Conformal Maps and Minimal Surfaces