Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

We will study metric measure spaces $$(X,\mathsf{d},{\mathfrak {m}})$$ ( X , d , m ) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds $$\mathsf{BE}_1(\kappa ,\infty )$$ BE 1 ( κ , ∞ ) for which we prove the equivalence with sharp gradient estimates, the class of which will be preserved under time changes with arbitrary $$\psi \in \mathrm {Lip}_b(X)$$ ψ ∈ Lip b ( X ) , and which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $$Y\subset X$$ Y ⊂ X . In the latter case, the distribution-valued Ricci bound will be given by the signed measure $$\kappa = k\,{\mathfrak {m}}_Y + \ell \,\sigma _{\partial Y}$$ κ = k m Y + ℓ σ ∂ Y where k denotes a variable synthetic lower bound for the Ricci curvature of X and $$\ell $$ ℓ denotes a lower bound for the “curvature of the boundary” of Y, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.

Wave emission from heterogeneities with changing boundary curvature and orientation by a circularly polarized electric field in cardiac tissues

In hearts, complex spatial–temporal patterns of action potential waves may cause life-threatening arrhythmia. Unlike the conventional defibrillation which uses high-voltage electric shocks associated with severe side effects, the new method of wave emission from heterogeneities (WEHs) merits close investigation. In our previous studies of the WEH to terminate arrhythmia in idealized conditions, we found that a circularly polarized electric field (CPEF) not only needs a lower voltage, but also has higher efficiency than a uniform electric field (UEF). But the effect of a CPEF on a real cardiac heterogeneity with irregular boundary shape remains unknown. Here, we consider elliptical heterogeneities whose boundary curvatures and orientations change in a similar way as irregular heterogeneities and study the effect of the changing boundary curvature and orientation on the WEH. We find that, unlike the UEF, the CPEF is not affected by the change of boundary curvature and orientation. Besides, the CPEF needs a lower voltage to induce wave emission from an elliptical heterogeneity than the UEF. Hence, it has advantages for the application of the WEH in clinical treatments.

Scale-invariant singularity of the surface quasigeostrophic patch

Numerical simulations of the surface quasigeostrophic patch indicate the development of a scale-invariant singularity of the boundary curvature in finite time, with some evidence of universality across a variety of initial conditions. At the time of singularity, boundary segments are shown to possess an exact and simple analytic form, described by branches of a logarithmic spiral centred on the point of singularity. The angles between the branches depend non-trivially on the shape of the smooth connecting boundary as the singularity is approached, but are independent of the global boundary.