https://neckpinch.com/authors/julius-baldauf/New arXiv papers by Julius Baldauf, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2210.14286arxiv:2210.14286Tue, 25 Oct 2022 00:00:00 GMTJulius Baldauf, Tang-Kai Lee — 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.Mean curvature flowStability & uniqueness https://arxiv.org/abs/2210.14198arxiv:2210.14198Tue, 25 Oct 2022 00:00:00 GMTJulius Baldauf — 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.Ricci flowRicci-flat & Calabi–Yau4-manifolds https://arxiv.org/abs/2206.09198arxiv:2206.09198Sat, 18 Jun 2022 00:00:00 GMTJulius Baldauf, Tristan Ozuch — 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.Ricci flowBoundaries & non-compactMinimal & CMC surfaces https://arxiv.org/abs/2201.05505arxiv:2201.05505Fri, 14 Jan 2022 00:00:00 GMTJulius Baldauf, Dain Kim — 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.Ricci flowStability & uniqueness https://arxiv.org/abs/2201.04475arxiv:2201.04475Wed, 12 Jan 2022 00:00:00 GMTJulius Baldauf, Tristan Ozuch — This paper generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) 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, on such manifolds, 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.Ricci flowScalar curvatureEigenvalues & spectrumPhysics