Calculating the inverse of an automorphism of a formal power series ring presents a frequent challenge in a myriad of mathematical inquiries, especially in the realm of singularity theory. In instances involving non-linear and multivariable contexts, S. S. Abhyankar pioneered a methodology to tackle this problem. However, calculating the expressions up to a certain order using this method requires calculating higher-order terms and then carry out the selection, which leads to redundant computations in practice. This article introduces two novel approaches for determining the inverse of an automorphism of a formal power series ring over an arbitrary commutative ring with unit, grounded in the newly developed higher order Jacobian matrix theory. These approaches can be conceived as non-linear extensions of the inverse matrix method and the Gaussian elimination method respectively. They avoid redundant computations above. For the two new methods, we also give the application in calculating the explicit expression for the implicit function theorem.
We present two characterizations of smooth compact Ricci flow solutions solely in terms of metrics and measures (one of them only works under positive scalar curvature along the flow); thus, provide weak formulations that are generalized to the singular setting in a straightforward manner. These formulations are achieved by weakly formulating super Ricci flows and imposing a saturation condition (solely in terms of metric and measure) to ensure the super Ricci flow inequality is an equality.
We prove the local Lipschitz continuity of viscosity solutions for two-phase free boundary problems for the p-Laplacian with nonzero right hand side, where p is an element of (1, infinity). This is the optimal regularity for the problem. We also obtain the local Ho & uml;lder continuity for a larger class of problems. The results introduced here are new even in the homogeneous situation, that is, when the right hand side is zero. Our work applies to merely viscosity solutions, which allows a wide applicability.
We prove the existence and the 1/2-Hölder continuity in time of flat flows for periodic Lipschitz subgraphs, whose evolution is governed by the gradient flow of generalized nonlocal perimeters. Moreover, we show that the flat flow satisfies the semigroup property and, as a consequence, the generalized perimeter decreases along the evolution. Finally, we prove that halfspaces are global minimizers of the generalized nonlocal perimeters and act as attractors for the dynamics. Our theory covers several generalized perimeters, including fractional and Riesz-type perimeters (defined on entire periodic subgraphs through suitable renormalization procedures) and the Minkowski pre-content.
Area-constrained critical surfaces for the Hawking quasi-local energy ("Hawking surfaces") provide a natural setting for that energy: they enjoy positivity and rigidity properties, as shown in earlier work [31]. We construct large-scale foliations at infinity by Hawking surfaces in asymptotically Schwarzschild initial data sets. Using a Lyapunov-Schmidt reduction within the Willmore foliation framework of Eichmair-Koerber [13], we prove existence and uniqueness of the foliation and study its coordinate center. Under the dominant energy condition, we show that along the leaves of the foliation, the Hawking energy is positive and converges to the ADM energy in the large-sphere limit; moreover, subject to an explicit integral constraint, it is monotone along the foliation. Under weaker assumptions we construct an on-center family of Hawking surfaces that, while not necessarily a foliation, still enjoys positivity and the large-sphere limit. Finally, we obtain a rigidity statement and verify that our hypotheses hold in a broad class of data, initial data sets with harmonic or York asymptotics, thereby demonstrating the robustness of Hawking surfaces as a quasi-local energy tool in dynamical spacetimes.
Improving a singularity theorem in General Relativity by Galloway and Ling we show the following (cf. Theorem 1): If a globally hyperbolic spacetime M satisfying the null energy condition contains a closed, spacelike Cauchy surface (V, g, K) (with metric g and extrinsic curvature K) which is 2-convex (meaning that the sum of the lowest two eigenvalues of K is non-negative), then either M is past null geodesically incomplete, or V is a spherical space, or V or some finite cover V is a surface bundle over the circle, with totally geodesic fibers. Moreover, (cf. Theorem 2) if (V, g, K) admits a U(1) isometry group with corresponding Killing vector , we can relax the convexity requirement in terms of a decomposition of K with respect to the directions parallel and orthogonal to . Finally, (cf. Propositions 1-3) in the special cases that V is either nonorientable, or non-prime, or an orientable Haken manifold with vanishing second homology, we obtain stronger statements in both Theorems without passing to covers. While our results do not use Einstein's equations, we provide several classes of Lambda-vacuum solutions of these equations as examples.
This is a continuation of our earlier work [14] on the Monge-Ampère obstacle problem D^2 v = v^q χ_{v>0}, v ≥ 0 convex with q ∈ [0,n), where we studied the regularity of the strictly convex part of the free boundary. In this work, we examine the non-strictly convex part of the free boundary and establish optimal dimension bounds for its flat portion. Additionally, we investigate the strong maximum principle and a stability property for this Monge-Ampère obstacle problem.
Nearly G_2-structures define positive Einstein metrics in 7 dimensions and are critical points, up to scale, for a geometric flow of co-closed G_2-structures with good analytic properties called the modified G_2-Laplacian co-flow. We introduce a suitable normalization of this flow so that nearly G_2-structures are stable under rescaling. However, we show that many nearly G_2-structures are unstable for this flow: specifically, all those naturally arising from 3-Sasakian geometry. In particular, we demonstrate that the standard nearly G_2-structure on the round 7-sphere is an unstable critical point with high index.
We explore novel properties of the biharmonic heat kernel on Euclidean space and derive an entropy type quantity for the extrinsic biharmonic map heat flow which exhibits monotonicity behaviors for n≤ 4.
In this paper, we prove that any compact 2-sided smooth stable minimal hypersurface in a shrinking gradient Ricci soliton (M-n, g, f) with scalar curvature R >= (n-1)lambda must have vanished second fundamental form and vanished normal Ricci curvature. For shrinking gradient Ricci solitons with scalar curvature R >= (n-1)lambda, the existence of an area-minimizing hypersurface would imply that M is splitting.
We obtain existence and local uniqueness of asymptotically flat, static vacuum extensions for Bartnik data on a sphere near the data of a sphere of symmetry in a Schwarzschild manifold.
We give examples of spin $4$-manifolds with boundary that do not admit metrics of positive scalar curvature and nonnegative mean curvature. These manifolds in fact have the stronger property that the conformal Laplacian with appropriate boundary conditions is never positive. The obstruction to the positivity of the conformal Laplacian is given by a real-valued $\xi$-invariant associated to the APS theorem for the twisted Dirac operator. We use analytic techniques related to the prescribed scalar curvature problem in conformal geometry to move beyond earlier work where the metric is a product near the boundary.
We provide new examples of 3-manifolds with weight one fundamental group and the same integral homology as the lens space L(2k,1) which are not surgery on any knot in the three-sphere. Our argument uses Furuta's 10/8-theorem, and is simple and combinatorial to apply.
Green's inequality shows that a compact Riemannian manifold with scalar curvature at least n(n - 1) has injectivity radius at most pi, and that equality is achieved only for the radius 1 sphere. In this work we show how extra topological assumptions can lead to stronger upper bounds. The topologies we consider are positive scalar curvature except lens spaces L(p, q) with p odd. We also prove a strengthened inequality for 3-manifolds with positive scalar curvature and large diameter. Our proof uses previous results of Gromov and Zhu.
We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical L^p-integrable, this flow converges locally smoothly to a limiting metric g(∞) on M with (M,g(∞)) isometric to the standard flat ℝ^n, which implies topological rigidity of M. This generalizes work of Chen , who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above.
We prove a transverse diameter theorem in the context of Lorentzian foliations, which can be interpreted as a Hawking–Penrose-type singularity theorem for timelike geodesics transverse to the foliation. In order to develop the necessary machinery we introduce and study a novel causality structure on the leaf space via the transverse Lorentzian geometry on the foliated manifold. We describe the initial rungs of a transverse causal ladder and relate them to their standard counterparts on an underlying foliated spacetime. We show how these results can be interpreted as doing Lorentzian (and more generally semi-Riemannian) geometry on low-regularity spaces that can be realized as leaf spaces of foliations. Accordingly, we discuss how all of these concepts and results apply to Lorentzian orbifolds, insofar as these can be seen as leaf spaces of a specific class of Lorentzian foliations. In particular, we derive an associated Lorentzian timelike diameter theorem on orbifolds.
We prove that the moduli space of mean convex two-spheres embedded in complete, orientable 3-dimensional Riemannian manifolds with nonnegative Ricci curvature is path-connected. This result is sharp in the sense that neither of the conditions of (strict) mean convexity, completeness, and nonnegativity of the Ricci curvature can be dropped or weakened. We also study the number of path components of mean convex Heegaard tori, again in ambient manifolds with nonnegative Ricci curvature. We prove that there are always either one or two path components and this number does not only depend on the homotopy type of the ambient manifold. We give a precise characterisation of the two cases and also discuss what happens if the mean convexity condition is weakened to nonnegative mean curvature.
The aim of this paper is to study almost rigidity properties of super Ricci flow whose Muller quantity is non-negative. We conclude almost splitting and quantitative stratification theorems that have been established by Bamler for Ricci flow. As a byproduct, we obtain an almost constancy for a certain integral quantity concerning scalar curvature at an almost selfsimilar point, which is new even for Ricci flow.
