Generalizes DeTurck's theorem to non-compact manifolds.
problem Uniquely determine Levi-Civita connection from Ricci curvature.
method Extends DeTurck's theorem to non-compact manifolds.
result Generalization to non-compact manifolds with finite total scalar curvature.
Study proves rigidity of non-negative scalar curvature on Euclidean space perturbations.
problem Rigidity of non-negative scalar curvature on Euclidean space perturbations.
method Analysis of long time solutions of Ricci DeTurck flow.
result Proves a weak version of the positive mass theorem.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Study Ricci-Deturck flow from rough metrics, proving short-time existence.
problem Short-time existence of Ricci-Deturck flow from rough metrics.
method Ricci-Deturck flow, bi-Lipschitz metrics, small gradient concentration.
result Proved short-time existence of Ricci-Deturck flow.
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.
New perspective on G2-structures flow from DeTurck Laplacian.
problem Understanding G2-structures and their flows.
method Introducing a new flow (DeTurck Laplacian flow) for G2-structures.
result DeTurck Laplacian flow is a flow of G2-structures.
Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.
problem Short-time existence of Ricci-DeTurck flow from rough metrics with specific integrability condition.
method Rough existence theory, preservation and improvement of scalar curvature bounds.
result Preservation and improvement of distributional scalar curvature lower bounds under certain conditions.
Defines mass for non-smooth hyperbolic spaces using a modified flow.
problem Defining mass for non-smooth, asymptotically hyperbolic spaces.
method Normalized Ricci-DeTurck flow with scalar curvature lower bound.
result Mass function well-defined for continuous metrics.
Study the Ricci flow on Finsler surfaces and find solutions.
problem Existence and uniqueness of solutions to the Ricci flow on Finsler surfaces.
method First, study the Finslerian Ricci-DeTurck flow and find a unique short time solution. Then, use this to find a solution to the original Ricci flow.
result Find solutions to the Ricci flow on Finsler surfaces.
Study compares volumes of hyperbolic 3-manifolds using Ricci-DeTurck flow.
problem Volume comparison on finite-volume hyperbolic 3-manifolds.
method Exponential convergence of Ricci-DeTurck flow to the hyperbolic metric.
result The hyperbolic metric minimizes volume among metrics with bounded scalar curvature.
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 C0 metrics that agrees with ADM mass.
problem Understanding ADM mass for C0 metrics and its behavior under Ricci-DeTurck flow. method Developed a C0 mass quantity and analyzed its behavior under Ricci-DeTurck flow. result The C0 mass at infinity is independent of coordinate charts and has controlled distortion under Ricci-DeTurck flow. Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. Simplified proof of stability for Ricci flow near ALE metrics.
problem Stability of Ricci flow near ALE metrics with integrable deformations.
method Equivalence between integrability and almost-orthogonality property of Ricci-DeTurck tensor, analysis in weighted Holder spaces.
result Dynamical stability of Ricci flow near linearly stable Ricci-flat ALE metrics.
Harmonic gauge simplifies geometric analysis of Riemannian metrics.
problem Analyzing the Hilbert-Einstein functional and its stability.
method Developed a harmonic gauge to eliminate divergence terms and induce elliptic structure.
result Positivity of curvature operator implies spectral stability of the functional.
The paper discusses a flow for almost continuous metrics with bounded curvature, leading to smooth metrics with bounded scalar curvature.
problem Riemannian manifolds with almost continuous metrics and bounded curvature.
method Ricci-DeTurck flow applied to (1−ε0(n))h≤g0≤(1+ε0(n))h result Smooth metrics with bounded scalar curvature can be obtained from almost continuous metrics.
New topological quantum gravity theories linked to Ricci flow.
problem Quantum gravity and geometric flows on manifolds.
method BRST quantization, gauging symmetries, localization.
result Path integral localized to Ricci flow solutions.
The paper establishes bounds on scalar curvature on asymptotically flat manifolds.
problem Establishing scalar curvature bounds on asymptotically flat manifolds.
method Using Ricci-DeTurck flow and distributional scalar curvature, the paper derives bounds on scalar curvature.
result The scalar curvature lower bound under Ricci-DeTurck flow depends on the scalar curvature lower bound in the β-weak sense and time.
This study proves the local existence of a symplectic gradient flow on a flat torus.
problem Proving the local existence of a symplectic gradient flow on a flat torus.
method Using a moment map and a DeTurck trick to make the flow strictly parabolic and showing local existence and regularity.
result The group of symplectomorphisms of the real four-dimensional torus is locally contractible.
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.
problem Approximating anisotropic curve shortening flow.
method Weak formulation and finite element approximation.
result Optimal H1-error bound for approximation. 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…
A new geometric flow K-flow on 3-manifolds shrinks or preserves homogeneous spheres.
problem Analyzing the behavior of Thurston's model geometries under the K-flow. method Defining and studying the K-flow on 3-dimensional Riemannian manifolds, using a DeTurck-type argument for short-time existence. result The K-flow shrinks or preserves homogeneous spheres, showing short-time existence. 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…
Study polyhomogeneity of metrics with Lie structure along Ricci flow.
problem Polyhomogeneity of metrics with Lie structure along Ricci flow.
method Analyzing polyhomogeneity of complete Riemannian metrics with Lie structure fibered at infinity under Ricci flow.
result Polyhomogeneity of metrics compatible with a Lie structure fibered at infinity is locally preserved by the Ricci-DeTurck flow.
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.…
Karigiannis discusses geometric flows of G2-structures, focusing on existence and uniqueness.
problem Existence and uniqueness of geometric flows of G2-structures.
method Introduced geometric structures, geometric flows, and discussed qualitative features. Focused on Ricci flow and DeTurck trick, then extended to G2-structures.
result Clarified conditions for short-time existence and uniqueness of G2-Laplacian flow.
The paper constructs Ricci flow solutions for non-smooth metrics in four dimensions.
problem Constructing Ricci flow solutions for non-smooth metrics in four dimensions.
method Ricci DeTurck flow on closed manifolds with initial values in W2,2. result Constructs solutions to Ricci flow for non-smooth metrics in four dimensions.
Uniform proof for Ricci flows on complete manifolds.
problem Proving short-time existence, uniqueness, and continuous dependence for Ricci flows.
method Using Koch-Lamm framework and tensor heat kernel estimates, with a new continuous dependence estimate.
result Uniform proof of short-time existence, uniqueness, and continuous dependence for Ricci flows.
It is the purpose of this article to establish a technical tool to study regularity of solutions to parabolic equations on manifolds. As applications of this technique, we prove that solutions to the Ricci-DeTurck flow, the surface diffusion flow and the mean curvature flow enjoy joint analyticity in time and space, an…
Extends cohomology to incomplete Riemannian manifolds.
problem Cohomology of harmonic forms on incomplete Riemannian manifolds.
method Abstract setting of Hilbert complexes.
result Geometric applications to Thom-Mather spaces.
Smooth Ricci flows from metric spaces can be extended to smooth solutions.
problem Extending Ricci flows from incomplete metric spaces to complete solutions.
method Ricci-harmonic map heat flow and δ-Ricci-DeTurck flow on Euclidean balls.
result Original Ricci flow can be extended to a smooth solution on a larger ball.
We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial …
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…
The article proves Kähler 4-manifolds can only be products of 2 surfaces.
problem Finding orthogonal coordinates on Kähler manifolds.
method Using geometric and algebraic methods to construct orthogonal coordinates.
result Only Kähler 4-manifolds can be products of 2 surfaces.
We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence of a global unique and analytic solution to the Ricci-DeTurck flow on euclidean s…
Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, existence of solutions to the so called Hamilton Ricci flow on Finsler spaces is studied and a short time solution is found. To this end the Finslerian Ricci-DeTurck flow on Finsle…
Extends optimal regularity and compactness to vector bundles over non-Riemannian manifolds.
problem Optimal regularity and compactness for connections on vector bundles.
method Derive RT-equations, establish existence theory, handle curvature up to L1. result Optimal regularity and compactness extended to vector bundles over non-Riemannian manifolds.
Modified Perelman entropy proves RG-2 flow monotonicity.
problem Analyzing RG-2 flow on Riemannian manifolds.
method Developed geometrically defined coupling constant and modified Perelman entropy.
result Modified Perelman entropy is monotonic under RG-2 flow.
Complex and quaternionic projective spaces lack local orthogonal coordinates.
problem Lack of local orthogonal coordinates in complex and quaternionic projective spaces.
method Analysis of Riemannian manifolds and canonical metrics.
result Complex and quaternionic projective spaces do not have local systems of orthogonal coordinates.
Study proves stability of slowly rotating Kerr black holes.
problem Stability of slowly rotating Kerr black holes.
method Linear stability analysis using wave map/DeTurck gauge, microlocal analysis, and non-elliptic Fredholm theory.
result Linearized perturbations decay at an inverse polynomial rate.
Ricci flow stability on manifolds with bounded geometry ensures convergence to hyperbolic metrics.
problem Stability and convergence of Ricci flow on manifolds with bounded geometry.
method Continuous dependence on initial conditions, sectoriality of Ricci-DeTurck flow generator, and Hölder norm analysis.
result Ricci flow converges to hyperbolic metrics under certain conditions.
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…
Higher-dimensional Ricci flows are shown to have unique and stable solutions.
problem Stability and uniqueness of Ricci flows in higher dimensions.
method Generalization of Bamler-Kleiner's proof to higher dimensions, use of Brendle's classification of κ-solutions, and maximum principle for linearized Ricci-DeTurck flow.
result Canonical evolution through singularities for manifolds with positive isotropic curvature.
Ricci flow stabilizes hyperbolic 3-manifolds near the hyperbolic metric.
problem Stability of Ricci flow on hyperbolic 3-manifolds.
method Normalized Ricci-DeTurck flow with exponential convergence to the hyperbolic metric.
result Normalized Ricci-DeTurck flow converges exponentially to the hyperbolic metric.
New constraints rule out some optimal domains for helicity maximisation.
problem Finding a smooth domain of fixed volume that maximizes helicity.
method Established additional geometric constraints on optimal domains.
result Ruled out the optimality of a broad class of solid tori.
Study geometric flows of G2-structures, determining curvature and torsion invariants.
problem Investigate geometric flows of G2-structures and their invariants.
method Explicitly compute differential invariants, decompose curvature and torsion, analyze principal symbols.
result Established short-time existence and uniqueness for geometric flows of G2-structures.
We consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we suppose G is a torus acting by isometries on M. Given X in the Lie algebra and corresponding vector field XM on M, one defines Witten's inhomogeneous coboundary operator dXM=d+ιXM:ΩG±→ΩG∓…