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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 May 15, 2023ArXivblowup, 奇点, 曲线流, 渐近性, 空间曲线 Leave a comment

Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow

Title: Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow Authors: Gabriel Khan Categories: math.DG math.AP Comments: 10 pages MSC-class: 53E10 53A04 \\ We study curve-shortening flow for twisted curves in $\mathbb{R}^3$ (i.e., curves with nowhere vanishing curvature $\kappa$ and torsion $\tau$) and define a notion of torsion-curvature entropy. Using this functional, we show that either the curve develops an inflection point or the eventual singularity is highly irregular (and likely impossible). In particular, it must be a Type II … Continue reading “Curvature-Torsion Entropy for Twisted Curves under Curve Shortening Flow”

Van Abel May 10, 2023StatusArXivblowup, conformal heat flow, evolution of metric, finite time singularity Leave a comment

Bubbling analysis of a conformal heat flow for…

Bubbling analysis of a conformal heat flow for harmonic maps Woongbae Park We study a conformal heat flow for harmonic maps. It is known that global weak solution of the flow exists and smooth except at mostly finitely many singular points. In this paper, we conduct a bubbling analysis for a finite time singularity.

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…”

Van Abel May 10, 2023StatusArXivblowup, expansion of connection forms, Yang-Mills Leave a comment

ON THE BLOW-UP OF YANG-MILLS FIELDS IN DIMENSION FOUR

ON THE BLOW-UP OF YANG-MILLS FIELDS IN DIMENSION FOUR In this paper, we study the blow-up of a sequence of Yang-Mills connection with bounded energy on a four manifold. We prove a set of equations relating the geometry of the bubble connection at the infinity with the geometry of the limit connection at the energy concentration point. These equations exclude certain scenarios from happening, for example, there is no sequence of Yang-Mills SU(2) connections on S4 converging to an ASD … Continue reading “ON THE BLOW-UP OF YANG-MILLS FIELDS IN DIMENSION FOUR”