MUSTREAD – MathRlife

Van Abel June 8, 2023ArXivAllard's regularity theorem, blowup, Brakke's regularity theorem, monotonicity formulas, MUSTREAD, varifolds, viscosity approach Leave a comment

A short proof of Allard’s and Brakke’s regularity theorems

Title: A short proof of Allard’s and Brakke’s regularity theorems Authors: Guido De Philippis, Carlo Gasparetto, Felix Schulze Categories: math.AP math.DG \\ We give new short proofs of Allard’s regularity theorem for varifolds with bounded first variation and Brakke’s regularity theorem for integral Brakke flows with bounded forcing. They are based on a decay of flatness, following from weighted versions of the respective monotonicity formulas, together with a characterization of non-homogeneous blow-ups using the viscosity approach introduced by Savin. \\ … Continue reading “A short proof of Allard’s and Brakke’s regularity theorems”

Van Abel June 1, 2023ArXiv$\Phi_{(3)}$-conservation law, $\phi_3$-harmonic maps, extrinsic average variational method, first variation formula, Liouville type, monotonicity formula, MUSTREAD, second variation formula, sphere theorems, stress-energy tensor, topological vanishing theorems, Unstable manifolds Leave a comment

The geometry of $\Phi_{(3)}$-harmonic maps

Title: The geometry of $\Phi_{(3)}$-harmonic maps Authors: Shuxiang Feng, Yingbo Han, Kaige Jiang and Shihshu Walter Wei Categories: math.DG math-ph math.AP math.MP Comments: 46 pages, to appear in Nonlinear Analysis (2023). arXiv admin note: text overlap with arXiv:1911.05855 MSC-class: 58E20, 53C21, 53C25 \\ In this paper, we motivate and extend the study of harmonic maps or $\Phi_{(1)}$-harmonic maps (cf [15], Remark 1.3 (iii)), $\Phi$-harmonic maps or $\Phi_{(2)}$-harmonic maps (cf. [24], Remark 1.3 (v)), and explore geometric properties of $\Phi_{(3)}$-harmonic maps … Continue reading “The geometry of $\Phi_{(3)}$-harmonic maps”

Van Abel May 10, 2023StatusArXivAbelian Yang-Mills-Higgs, asymptotic behaviour, compactness, monotonicity formula, MUSTREAD, rectifiable varifold Leave a comment

The Yang Mills Higgs functional on complex line…

The Yang-Mills-Higgs functional on complex line bundles: asymptotics for critical points Giacomo Canevari, Federico Luigi Dipasquale, Giandomenico Orlandi We consider a gauge-invariant Ginzburg-Landau functional (also known as Abelian Yang-Mills-Higgs model) on Hermitian line bundles over closed Riemannian manifolds of dimension n≥3. Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the non-self dual scaling, as the coupling parameter tends to zero. After a convenient choice of the gauge, we show compactness … Continue reading “The Yang Mills Higgs functional on complex line…”

Van Abel May 10, 2023StatusArXivany time sequence, blowup, continuous in time, harmonic maps, MUSTREAD Leave a comment

Continuous in time bubble decomposition for the harmonic…

Continuous in time bubble decomposition for the harmonic map heat flow Jacek Jendrej, Andrew Lawrie, Wilhelm Schlag We consider the harmonic map heat flow for maps from the plane to the two-sphere. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches … Continue reading “Continuous in time bubble decomposition for the harmonic…”