Complete noncompact solitons split at infinity under certain conditions.
problem Conditions for complete noncompact shrinking gradient Ricci solitons to split at infinity.
method Geometric conditions analysis.
result Conditions under which complete noncompact solitons split at infinity.
The paper splits manifolds using infinity harmonic functions with linear growth.
problem Splitting manifolds with specific harmonic functions.
method Analyzes manifolds with non-negative Ricci or sectional curvature, focusing on infinity harmonic functions with linear growth.
result Extends Savin's theorem to surfaces with non-negative sectional curvature.
Introduces semistability in geometric group theory and provides techniques to prove it.
problem Whether all finitely presented groups are semistable at infinity.
method Techniques involving the topology of the boundary or hierarchies of splittings.
result Illustrates semistability for hyperbolic relative groups using hierarchies of splittings.
The study proves manifolds with specific curvature and volume properties always split off a line at infinity.
problem Understanding the geometry at infinity of manifolds with linear volume growth and nonnegative Ricci curvature.
method Analyzing properties of Busemann functions and constructing examples.
result Manifolds with the specified properties always split off a line at infinity, with bounded diameter of level sets of Busemann functions.
We generalize the Bartsch-Li's splitting lemma at infinity for C2-functionals in [2] and some later variants of it to a class of continuously directional differentiable functionals on Hilbert spaces. Different from the previous flow methods our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-…
Study complete manifolds with weighted Poincaré inequality and Ricci curvature bounds.
problem Understanding the structure of complete manifolds with specific curvature and inequality conditions.
method Analyzing manifolds with weighted Poincaré inequality and Ricci curvature bounds.
result Obtained splitting results for manifolds with non-zero weight function limit at infinity.
Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
This research shows that steady solitons in higher dimensions always reduce at infinity.
problem Characterizing steady solitons with nonnegative sectional curvature in higher dimensions.
method Dimension reduction analysis and tangent flow classification.
result Steady solitons in higher dimensions always reduce at infinity.
New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
problem Proving splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
method New proof of splitting theorem and construction of weighted minimizing geodesics at infinity.
result Minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends.
Refined asymptotics of scalar-flat ALE four-manifolds
problem Asymptotic behavior of scalar-flat ALE four-manifolds
method Constructing preferred coordinates at infinity
result Identifying homogeneous ∣x∣−2 term in metric expansion The paper studies linearisation and splitting properties for vector fields and algebras.
problem Linearisation and splitting properties for vector fields and algebras.
method Formal linearisation problem in the framework of graded coalgebras, explicit recursive construction, and characterisation of linearisable algebras.
result A formal vector field is linearisable if and only if it satisfies a splitting property, providing a streamlined proof of Basto-Gonçalves' theorem.
CAT(0) spaces can split into products if their boundary has certain properties.
problem Detecting product splittings in CAT(0) spaces.
method Analyzing the boundary properties of CAT(0) spaces and their group actions.
result Conditions for a CAT(0) space to contain a quasi-dense, closed convex subspace that splits as a product.
Split Courant algebroids linked to special algebra structures.
problem Understanding the structure of split Courant algebroids.
method Established a correspondence with multiplicative curved L∞-algebras. result Split Courant algebroids correspond to multiplicative curved L∞-algebras. The paper examines perimeter minimizing sets in curved spaces and finds conditions for their boundary to match a specific structure.
problem Conditions for perimeter minimizing sets in curved spaces to have a boundary matching a product structure.
method Analyzes Riemannian manifolds with non-negative sectional curvature and quadratic volume growth.
result The boundary of a perimeter minimizing set in such manifolds is identified with a slice in the product structure.
The paper extends topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.
problem Generalizing topological results to noncompact spaces with nonnegative N-Bakry Émery Ricci curvature.
method Study of the Splitting Theorem and geodesic loops to infinity property in relation to spaces with nonnegative N-Bakry Émery Ricci curvature.
result If Mn is a complete, noncompact Riemannian manifold with nonnegative N-Bakry Émery Ricci curvature where N>n, then Hn−1(M,Z) is 0. Study shows hyperbolic links are not common among prime links.
problem Understanding the prevalence of hyperbolic links among prime links.
method Analyzing satellite and prime non-split links of varying complexity.
result The proportion of hyperbolic links among all prime non-split links does not approach 1 as complexity increases.
The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least C2-smooth functionals. In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower…
This paper concerns complete noncompact manifolds with nonnegative Ricci curvature. Roughly, we say that M has the loops to infinity property if given any noncontractible closed curve, C, and given any compact set, K, there exists a closed curve contained in M\K which is homotopic to C. The main theorems in this paper …
We review geometrical properties of a static spacetime (M,g), including geodesic completeness, causality, standard splittings, compact M, closed geodesics and geodesic connectedness. We pay special attention to the critical quadratic behavior at infinity of the coefficients β, β−1 (β=−g(K,K), being K a …
In a 3-manifold M, let K be a knot and R be an annulus which meets K transversely. We define the notion of the pair (R,K) being caught by a surface Q in the exterior of the link given by K and the boundary curves of R. For a caught pair (R,K), we consider the knot K^n gotten by twisting K n times along R and give a low…
The study proves splitting theorems for manifolds with specific curvature and boundary conditions.
problem Proving splitting theorems for manifolds with specific curvature and boundary conditions.
method Warped product splitting theorem in manifolds with Ricci curvature bounded from below, requiring parabolic and convex boundary.
result Established splitting results for various manifolds with specific curvature and boundary conditions.
We consider solutions to the anti-self-dual Yang Mills (ASDYM) equations in split signature that are global on the double cover of the appropriate conformally compactified Minkowski space $\widetilde\M$. Ward's ASDYM twistor construction is adapted to this geometry by using a correspondence between points of $\widetild…
Exact distribution of split conformal prediction coverage found.
problem Determining the reliability of prediction sets in batch mode.
method Analysis of exchangeable data to find universal distribution of empirical coverage.
result Exact distribution of empirical coverage is universal and determined by nominal miscoverage level and calibration sample size.
We study geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry-Emery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtai…
The study proves properties of capillary graphs in half-spaces.
problem Characterizing capillary minimal graphs in half-spaces.
method Analyzing tangent cones and regular set properties.
result Capillary minimal graphs in low dimensions or specific cone conditions are linear.
Study minimal graphs on non-negative Ricci curvature manifolds.
problem Minimal graphs with linear growth on manifolds with non-negative Ricci curvature.
method New gradient estimate for minimal graphs and heat equation techniques.
result Non-constant minimal graphs force tangent cones to split off a line.
The study finds an infinite number of minimal surfaces in 3D spheres.
problem Finding minimal surfaces in 3D spheres.
method Two-parameter min-max scheme in lens spaces, Heegaard foliations flipping.
result Constructs an infinite number of minimal surfaces in S3. 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.
We generalize the splitting theorem of Cai-Galloway for complete Riemannian manifolds with $\Ric\geq-(n-1)$ admitting a family of compact hypersurfaces tending to infinity with mean curvatures tending to n−1 sufficiently fast to the setting of smooth metric measure spaces. This result complements and provides a new p…
We study minimal graphic functions on complete Riemannian manifolds $\Si$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with …
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…
Paper uses Carleman estimates to study harmonic functions on surfaces at infinity.
problem Unique continuation of harmonic functions on surfaces at infinity.
method Develops Carleman estimates for surfaces in Euclidean space at infinity.
result Obtains unique continuation property for harmonic functions.
Study on bounded solutions of parabolic and elliptic equations on noncompact Riemannian manifolds with conditions at infinity.
problem Existence and uniqueness of bounded solutions for equations with unbounded coefficients.
method Investigation of solutions satisfying prescribed conditions at infinity, considering large-time behavior.
result Established existence of solutions for parabolic and elliptic equations on noncompact Riemannian manifolds.
The paper examines mass aspects at future null infinity and limits of quasilocal mass.
problem Understanding mass aspects and limits of quasilocal mass at future null infinity.
method Review and extension of Bondi mass and mass loss formula in Bondi-Sachs coordinate system.
result New results about the limit of quasilocal mass of unit spheres at null infinity.
The paper defines curvature at infinity for flat manifolds.
problem Defining curvature at the boundary of flat manifolds.
method Constructing coordinates at infinity for asymptotically flat ends.
result A Weyl tensor and renormalized volume defined at infinity.
Proves unique continuation at infinity for certain expanding Ricci solitons.
problem Unique continuation at the boundary of expanding Ricci solitons.
method Establishes Carleman inequalities for weighted Laplacian.
result Unique continuation at infinity for asymptotically Ricci flat Ricci expanders.
The paper studies the structure at infinity of shrinking Ricci solitons with bounded curvature.
problem Understanding the structure at infinity of shrinking Ricci solitons with bounded curvature.
method Analyzes the limit behavior of complete gradient shrinking Ricci solitons and applies results to four-dimensional cases.
result The end of a shrinking Ricci soliton is asymptotic to a round cylinder at infinity.
Extends a theorem for first-order elliptic operators on manifolds.
problem Proving the relative index theorem for general first-order elliptic operators.
method Using boundary value problems and graphical decomposition of elliptically regular boundary conditions.
result Proves the relative index theorem for general first-order elliptic operators.
Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects…
We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.
Computes quasi-local mass at null infinity using Bondi-Sachs coordinates.
problem Global properties of quasi-local mass at null infinity.
method Evaluation of Wang-Yau quasi-local mass on unit spheres in Bondi-Sachs coordinates.
result Quasi-local mass is related to the news function in Bondi-Sachs coordinates.
The paper finds power series for Bach-flat metrics from spacetimes, including Einstein and constant curvature cases.
problem Extracting asymptotically anti-de Sitter Einstein 4-metrics from Bach-flat spacetimes.
method Using conformally compact Riemannian setting and formal power series, the paper finds expansions about conformal infinity.
result The mass is part of the free data at conformal infinity, leading to Einstein metrics.
Study asymptotic behavior of Weingarten surfaces at infinity.
problem Understanding the behavior of Weingarten surfaces at infinity.
method Derive asymptotic expansion and solve Dirichlet problem.
result Established maximum principle and solved Dirichlet problem.
Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
Riemannian manifolds can be sphere at infinity of certain solitons.
problem Understanding the structure of asymptotically conical expanding Ricci solitons.
method Formal expansions to show any compact manifold can be a sphere at infinity of a soliton.
result Any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.
We look at complete minimal surfaces of finite total curvature in R4. Similarly to the case of complex curves in C2 we introduce their {\it link at infinity}; we derive the {\it writhe number at infinity} which gives a formula for the total normal curvature of the surface. The knowledge of the l…
For a polynomial f in two complex variables, we prove that the multi-link at infinity of the 0-fiber f−1(0) is a fibred multi-link if and only if all the values different from 0 are regular at infinity.