Julius Baldauf

Publications & preprints

• Harmonic spinors in the Ricci flow. (2022)
  arXiv

  Abstract

  This paper studies the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.

• Parabolic frequency for the mean curvature flow. (2022)
  arXiv (with Tang-Kai Lee)

  Abstract

  This paper defines a parabolic frequency for solutions of the heat equation along homothetically shrinking mean curvature flows and proves its monotonicity along such flows. As a corollary, frequency monotonicity provides a proof of backwards uniqueness. Additionally, for solutions of more general parabolic equations on mean curvature flow shrinkers, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.

• The spinorial energy for asymptotically Euclidean Ricci flow. (2022)
  To appear in Advanced Nonlinear Studies.
  arXiv (with Tristan Ozuch)

  Abstract

  This paper introduces a functional generalizing Perelman's weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.

• Parabolic frequency on Ricci flows. (2022)
  International Mathematics Research Notices.
  arXiv, journal (with Dain Kim)

  Abstract

  This paper defines a parabolic frequency for solutions of the heat equation on a Ricci flow and proves it's monotonicity along the flow. Frequency monotonicity is known to have many useful consequences; here it is shown to provide a simple proof of backwards uniqueness. For solutions of more general parabolic equations on a Ricci flow, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.

• Spinors and mass on weighted manifolds. (2022)
  Communications in Mathematical Physics.
  arXiv, journal (with Tristan Ozuch)

  Abstract

  This paper generalizes classical spin geometry to the setting of weighted manifolds and provides applications to the Ricci flow. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, it is shown that on AE manifolds with nonnegative scalar curvature, Ricci flow is the gradient flow of said weighted ADM mass, for a natural choice of weight function. This yields a monotonicity formula for the weighted spinorial Dirichlet energy of a weighted Witten spinor along Ricci flow.

• Sharp entropy bounds for plane curves and dynamics of the curve shortening flow. (2018)
  To appear in Communications in Analysis and Geometry.
  arXiv (with Ao Sun)

  Abstract

  We prove that a closed immersed plane curve with total curvature 2πm has entropy at least m times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature 2πm whose entropy is less than m times the entropy of the embedded circle. As an application, we extend Colding-Minicozzi's notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.