Study fundamental groups of manifolds with nonnegative Ricci curvature and stability at infinity.
problem Understanding the fundamental groups of manifolds with nonnegative Ricci curvature and stability conditions.
method Analyzing tangent cones of the Riemannian universal cover and using Gromov-Hausdorff distance.
result Fundamental groups of manifolds with certain properties are finitely generated and contain abelian subgroups.
We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.
We study the nonlinear stability of the (3+1)-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the nonlinear initial value problem using an iteration scheme in which we solve a linea…
Proves conditions for generating families on Lagrangian cobordisms.
problem Existence of generating families on Lagrangian cobordisms.
method Analyzes exact Lagrangian cobordisms and Legendrian submanifolds using generating families.
result Conditions for extending generating families linear at infinity.
We study stability properties of f-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the top…
We study the fundamental group of an open n-manifold M of nonnegative Ricci curvature. We show that if there is an integer k such that any tangent cone at infinity of the Riemannian universal cover of M is a metric cone, whose maximal Euclidean factor has dimension k, then π1(M) is finitely generated. In p…
We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates.
Classifies stability of flat-core p-elasticae pinned at boundaries.
problem Stability of flat-core p-elasticae under pinned boundary conditions. method Classification based on previous work for all p∈(1,∞) and d≥2. result Completes the classification of stable pinned p-elasticae in Rd. We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large…
We study the asymptotic behavior of quantized Ding functionals along Bergman geodesic rays and prove that the slope at infinity can be expressed in terms of Donaldson-Futaki invariants and Chow weights. Based on the slope formula, we introduce a new algebro-geometric stability on Fano manifolds and show that the existe…
Paper addresses LSTM stability for thermal systems using infinity-norm.
problem Stability of LSTM networks in thermal systems.
method Derived ISS∞ condition for LSTM, developed training strategy. result ISS∞-promoted LSTM outperforms other models in thermal system case study. Novel approach to wave equations near null infinity in flat spacetimes.
problem Analyzing regularity and decay of wave equations near null infinity in asymptotically flat spacetimes.
method Microlocal analysis in a compactified spacetime with corners, focusing on edge-type wave operators.
result Microlocal regularity propagates across null infinity via radial sets, leading to new estimates for wave equations.
New proof of stability for expanding Kerr-de Sitter spacetimes with smoothness at the boundary.
problem Stability of expanding region of Kerr-de Sitter spacetimes.
method Modified generalized harmonic gauge, local stability near conformal boundary, smoothness down to future conformal boundary.
result Smoothness of conformally rescaled metric down to future conformal boundary with mild singularity.
Classifies and analyzes the stability of black hole event horizon birth points using contact geometry.
problem Classifying and understanding the structural possibilities of black hole crease sets.
method Contact geometry approach, focusing on BigFronts and their Legendrian projections.
result Refined stability discussion of the event horizon birth component and identification of additional components.
Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as…
Uniqueness proven for stable hypersurface tangent cones.
problem Stability and uniqueness of tangent cones for stable hypersurfaces.
method Analysis of isolated singularities and tangent cones of stable minimal hypersurfaces.
result Uniqueness of tangent cones with integer multiplicities.
Knot Floer homology stabilizes with twists.
problem Understanding knot behavior through homology.
method Analyzing m full twists on n parallel strands. result Knot Floer homologies stabilize as twists increase.
The paper studies optical properties in de Sitter spacetimes.
problem Stability of expanding regions in Schwarzschild de Sitter spacetimes.
method Construction of global solutions to the eikonal equation and detailed estimates of structure coefficients.
result Globally well-behaved double null foliations can be constructed from infinity.
This paper studies the normalized Ricci flow from a slight perturbation of the hyperbolic metric on Hn. It's proved that if the perturbation is small and decays sufficiently fast at the infinity, then the flow will converge exponentially fast to the hyperbolic metric when the dimension n>5.
We introduce a holomorphic sheaf E on a Sasaki manifold and study two new notions of stability for E along the Sasaki-Ricci flow related to the `jumping up' of the number of global holomorphic sections of E at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bound…
New Ricci flows found with Einstein orbifolds at infinity.
problem Ancient and immortal Ricci flows with Einstein orbifolds at infinity.
method Continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows constructed.
result Found continuous families of non-isometric ancient Ricci flows and half-PIC ancient flows on specific manifolds.
This note is meant to introduce the reader to a duality principle for nonlinear equations that recently appeared in the literature. Motivations come from the desire to give a unifying potential-theoretic framework for various maximum principles at infinity appearing in the literature (Ekeland, Omori-Yau, Pigola-Rigoli-…
Stability and rigidity results for Anosov flows on manifolds.
problem Stability and rigidity of Anosov flows and boundary actions.
method Topological stability and rigidity results using skew-Anosov flows.
result Construction of hyperbolic 3-manifolds with multiple topologically inequivalent Anosov flows.
The study examines stability and classification of special minimal hypersurfaces in high dimensions.
problem Stability and classification of special minimal hypersurfaces in high dimensions.
method Analysis of stability and nondegeneracy properties using Jacobi fields and Morse index.
result In high dimensions, these hypersurfaces are strictly stable and have a full classification of bounded Jacobi fields.
We prove homological stability for sequences of "oriented configuration spaces" as the number of points in the configuration goes to infinity. These are spaces of configurations of n points in a connected manifold M of dimension at least 2 which 'admits a boundary', with labels in a path-connected space X, and with an …
The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney C∞-topology). We show that a Morse function is stable if it is end-trivial at any point in its discriminant, where end-triviality (which is also called local triviality …
Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in R^3. It is shown that the unknot with maximal Thurston--Bennequin invariant of -1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant ge…
Builds geometric structures for algebraic groups over real closed fields.
problem Characterizing and decomposing algebraic groups over specific valued fields.
method Real algebraic geometry to construct and analyze affine buildings.
result Computed stabilizers and obtained group decompositions.
The paper shows how to stabilize perturbed Kähler-Ricci solitons.
problem Stabilizing perturbed Kähler-Ricci solitons.
method Normalized Kähler-Ricci flow starting from perturbed metrics.
result The flow converges to an asymptotically conical gradient expanding Kähler-Ricci soliton.
A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact symplectic manifolds. The notion of Eliashberg-Gromov convex ends provides a natural r…
Proves K-polystability for Kähler-Ricci shrinkers with decaying curvature.
problem K-stability of Kähler-Ricci shrinkers with decaying curvature.
method Developed algebraic theory for Kähler-Ricci shrinkers and proved K-polystability.
result Existence of Kähler-Ricci shrinker metric implies K-polystability in decaying curvature case.
A concrete model for a 7-dimensional gauge theory under special holonomy is proposed, within the paradigm outlined by Donaldson and Thomas, over the asymptotically cylindrical G2-manifolds provided by Kovalev's noncompact version of the Calabi conjecture. One obtains a solution to the G2-instanton equation from the …
New boundary condition for Black-Scholes equations in strict local martingale models.
problem Computing prices of European options with underlying asset as a strict local martingale.
method Numerical procedure using finite difference methods with a new boundary condition at infinity.
result The minimal solution, satisfying a discrete maximum principle, is the correct derivative price.
In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond to points in the Euclidean Weyl chamber. We can hence assign vector-valued side-lengths to segments. Our main result is a system of homogeneous linear inequalities describing the restrictions on the side -lengths of closed polygons. T…
A new definition of umbilic points at infinity for polynomial surfaces.
problem Defining umbilic points at infinity for homogeneous polynomial graphs.
method Proposed a stronger definition than Toponogov's, proving all are isolated and pairs, with geometric interpretation.
result All umbilic points at infinity are isolated and occur in pairs, being zeroes of the projective extension of the third fundamental form.
In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter $…
Proves convergence of PSGLA for sampling non-convex potentials.
problem Sampling from non-convex potentials with stability.
method Combines ULA and proximal optimization with stability analysis.
result First proof of convergence for PSGLA on non-convex potentials.
The study proves stability of the positive mass theorem for Kähler manifolds.
problem Stability of the positive mass theorem for Kähler manifolds.
method Integral inequality and stability results for ADM mass on AE Kähler manifolds.
result Stability of the positive mass theorem for Kähler manifolds under certain conditions.
We prove global existence for solutions arising from small initial data for a large class of quasilinear wave equations satisfying the `weak null condition' of Lindblad and Rodnianski, significantly enlarging upon the class of equations for which global existence is known. In addition to the usual weak null condition, …
Usually, in the Black-Scholes pricing theory the volatility is a positive real parameter. Here we explore what happens if it is allowed to be a complex number. The function for pricing a European option with a complex volatility has essential singularities at zero and infinity. The singularity at zero reflects the put-…
The paper proves mapping class groups of closed surfaces are simply connected at infinity.
problem Understanding connectivity at infinity for mapping class groups of surfaces.
method Proved a general simply connected at infinity result for finitely presented groups.
result All mapping class groups of closed surfaces of genus ≥ 3 are simply connected at infinity.
We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m at least 3, we construct an infinite family of pairs of m-manifolds on whic…
We continue the study of a general class of spaces of 0-cycles on a manifold defined and begun by Farb-Wolfson-Wood. Using work of Gadish on linear subspace arrangements, we obtain representation stability for the cohomology of the ordered version of these spaces. We establish subexponential bounds on the growth of uns…
Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact 3−folds, called building blocks, satisfying a stability condition `at infinity'. Such bundles are known to parametrise solutions of the Yang-Mills equation over the G2−manifolds obtain…
The paper extends results on minimal hypersurfaces in Riemannian manifolds to higher dimensions.
problem Characterizing properties of minimal hypersurfaces in higher-dimensional Riemannian manifolds.
method Maximum principle at infinity for two-sided, parabolic, properly embedded minimal hypersurfaces.
result Two disjoint properly embedded minimal hypersurfaces bound a slab in specific conditions.
The paper explores stability properties of cohomology groups and norms in symplectic and mapping class groups.
problem Stability properties of bounded cohomology in mapping class groups and symplectic groups.
method Utilizes results from Bestvina and Fujiwara, calculates norms of signature classes, and estimates cohomology norms.
result The bounded cohomology of mapping class groups does not stabilize, while that of symplectic groups does not stabilize via isometries.
Study shows minimizing the norm of the ERM solution stabilizes kernel ridge-less regression.
problem Stability of kernel ridge-less regression.
method Minimizing the norm of the ERM solution to minimize CV stability.
result Interpolating solution with minimum norm minimizes CV stability.
Stability proved for open Milne spacetime, showing gravity's long-term behavior.
problem Global stability of open Milne spacetime for Einstein-scalar field equations.
method Gaussian normal coordinates, exploiting expanding geometry of Milne spacetime.
result Spatial metric tends to hyperbolic metric as time goes to infinity.