arXiv research

This study proves the local existence of a symplectic gradient flow on a flat torus.

In this paper, we use the DeTurck trick to study the short-time existence of solutions to the Dirichlet and Newmann boundary problems of the cross curvature flow on 3-manifolds with boundary.

New method approximates anisotropic curve shortening flow.

We show that solutions to certain higher-order intrinsic geometric flows on a compact manifold, including some flows generated by the ambient obstruction tensor, are unique. With the goal of providing a complete self-contained proof, details surrounding map covariant derivatives and a careful application of the DeTurck…

Karigiannis discusses geometric flows of G2-structures, focusing on existence and uniqueness.

In this note we study conformal Ricci flow introduced by Arthur Fischer. We use DeTurck's trick to rewrite conformal Ricci flow as a strong parabolic-elliptic partial differential equations. Then we prove short time existences for conformal Ricci flow on compact manifolds as well as on asymptotically flat manifolds. We…

New topological quantum gravity theories linked to Ricci flow.

We demonstrate that the uniqueness of solutions to a broad class of parabolic geometric evolution equations can be proven via a direct and essentially classical energy argument which avoids the DeTurck trick entirely. Previously, we have used a variation of this technique to give an alternative proof and slight extensi…

Study geometric flows of G2-structures, determining curvature and torsion invariants.

Study Ricci-Deturck flow from rough metrics, proving short-time existence.

New perspective on G2-structures flow from DeTurck Laplacian.

Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.

Defines mass for non-smooth hyperbolic spaces using a modified flow.

We prove a rigidity result for non-negative scalar curvature perturbations of the Euclidean metric on $\mathbb{R^n}$ , which may be regarded as a weak version of the rigidity statement of the positive mass theorem. We prove our result by analyzing long time solutions of Ricci DeTurck flow. As a byproduct in doing so, w…

Study compares volumes of hyperbolic 3-manifolds using Ricci-DeTurck flow.

In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita…

We show that the polyhomogeneity at infinity of an asymptotically complex hyperbolic metric is preserved along the Ricci-DeTurck flow. Moreover, if the initial metric is `smooth up to the boundary', this will be preserved by the Ricci-DeTurck flow and the normalized Ricci flow. When the initial metric is Kähler, sharpe…

Study shows a mass quantity for $C^0$ metrics that agrees with ADM mass.

The paper classifies flows of SU(2)-structures on 4-manifolds.

Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to …

Smooths metrics with nonnegative scalar curvature near singular sets.

Volume comparison theorem for rank 1 symmetric spaces proved.

Harmonic gauge simplifies geometric analysis of Riemannian metrics.

The paper discusses a flow for almost continuous metrics with bounded curvature, leading to smooth metrics with bounded scalar curvature.

The paper establishes bounds on scalar curvature on asymptotically flat manifolds.

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon.

Study of skateboard flips as continuous curves in $SO(3)$ group.

Nash's theorem proved with Günther's trick

Perelman's Ricci flow emerges in quantum gravity, linking math and physics.

Explains Conway's tangle trick and its mathematical origins.

Unified framework for gradient estimation in combinatorial spaces.

We prove all knots can be transformed into a trefoil using special diagrams.

A new geometric flow $K$-flow on 3-manifolds shrinks or preserves homogeneous spheres.

Along the Ricci flow, we study the polyhomogeneity of complete Riemannian metrics endowed with "a Lie structure fibred at infinity", that is, a class of Lie structures at infinity that induce in a precise way a fibre bundle structure on a certain compactification by a manifold with corners. When the compactification is…

Geometric trick simplifies link homotopy and concordance.

In this paper, we study the relation of the monotonicity of Hawking Mass and geometric flow problems. We show that along the Hamilton-DeTurck flow with bounded curvature coupled with the modified mean curvature flow, the Hawking mass of the hypersphere with a sufficiently large radius in Schwarzschild spaces is monoton…

Outlier based Robust Principal Component Analysis (RPCA) requires centering of the non-outliers. We show a "bias trick" that automatically centers these non-outliers. Using this bias trick we obtain the first RPCA algorithm that is optimal with respect to centering.

The Gumbel-max trick and its extensions simplify sampling from categorical distributions in machine learning.

A new gradient estimator for categorical distributions reduces bias and variance.

Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.

Retail Product Image Classification is an important Computer Vision and Machine Learning problem for building real world systems like self-checkout stores and automated retail execution evaluation. In this work, we present various tricks to increase accuracy of Deep Learning models on different types of retail product …

Triple-point Whitney trick classifies ornaments of 3-manifolds.

Alexander trick applied to homology spheres for manifold homeomorphisms.

Embolic volume of compact manifolds is defined in terms of Berger's embolic inequality. In this paper, we show a result of relating embolic volume to the first Betti number. The proof relies on Gromov's covering argument appeared in systolic geometry. Berger called this method covering trick. We exploit and present mor…

Expands Bredon's trick for applications in geometry and topology.

Bredon's trick helps extend local properties to global topological spaces.

In this paper is considered the differential equation Ric(g)=T, where Ric(g) is the Ricci tensor of the metric g and T is a rotational symmetric tensor on R^n. A new, geometric, proof of the existence of smooth solutions of this equation, based on qualitative theory of implicitdifferential equations, is presented here.…

We observe that gradients computed via the reparameterization trick are in direct correspondence with solutions of the transport equation in the formalism of optimal transport. We use this perspective to compute (approximate) pathwise gradients for probability distributions not directly amenable to the reparameterizati…