Flat Higgs bundles restrict manifolds' curvature to be semi-negative.
problem Curvature restrictions on manifolds with Higgs bundles.
method Proving a semi-negative curvature property for manifolds with flat admissible Higgs bundles.
result Proved that curvature of manifolds with flat Higgs bundles is semi-negative.
New topological restrictions found for spaces with nonnegative Ricci curvature.
problem Understanding topological properties of spaces with nonnegative Ricci curvature.
method Analyzing complete Riemannian manifolds and RCD(0,n) spaces, applying rigidity and vanishing theorems.
result Proved a Betti number rigidity theorem and a vanishing theorem for simplicial volume.
In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann-Christoffel curvature tensor, the same type structure given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various ge…
The paper explores geometric structures related to projective curvature tensor.
problem Investigating geometric structures of projective curvature tensor.
method Study of semisymmetric and pseudosymmetric type curvature restricted geometric structures.
result Characterization and existence of geometric structures on Riemannian and semi-Riemannian manifolds.
New restrictions on holonomy groups for certain curvature conditions.
problem Restrictions on holonomy groups for Riemannian manifolds with specific curvature properties.
method Analyzing the curvature operator of the second kind to derive restrictions on holonomy groups.
result Holonomy groups are restricted to SO(n) or the manifold is flat for certain curvature conditions. Extends diameter bounds for submanifolds with boundary and minor curvature restrictions.
problem Bounding the diameter of submanifolds with boundary and minor curvature restrictions.
method Applies bounds dependent on mean curvature and area to minimal, constant mean curvature, and prescribed mean curvature surfaces.
result Diameter bounds for submanifolds with boundary and minor curvature restrictions.
New method for scalar curvature studies without manifold restrictions.
problem Scalar curvature constraints without manifold restrictions.
method Skin structures merging minimal hypersurface methods with surgery arguments.
result Obstruction and structure theories derived for scalar curvature constraints.
Study expands classical harmonic function results to Riemannian manifolds.
problem Classical harmonic function properties in domains of Riemannian manifolds.
method Generalized classical results to Riemannian manifolds, including pinched negative curvature.
result Generalized results for Riemannian manifolds, including pinched negative curvature.
Estimates spectral projections restricted to uniformly embedded submanifolds.
problem Estimating spectral projections on submanifolds of manifolds with nonpositive curvature.
method Estimates the L2(M)oLq(Σ) norm of spectral projection operators. result Sharp spectral projection estimates for small spectral windows.
The study restricts ancient, type-I, non-collapsing 2D mean curvature flows to spheres or cylinders.
problem Understanding blow-up limits of ancient solutions in mean curvature flow.
method Argument of Giga and Kohn to restrict flows to spheres or cylinders.
result Ancient, type-I, non-collapsing 2D mean curvature flows are restricted to spheres or cylinders.
Study on Dolbeault cohomology and Kähler foliations with new formulas and restrictions.
problem Investigating Dolbeault cohomology on transversely Kähler foliations.
method Developed new Weitzenböck formulas and conditions on curvature.
result Proved absence of nonzero basic-harmonic forms on foliations with positive Ricci curvature.
Study mean curvature forms on Kähler foliations with restrictions on basic cohomology.
problem Properties of mean curvature forms on transverse Kähler foliations.
method Analysis of mean curvature one-forms and their holomorphic/antiholomorphic counterparts.
result Restrictions on basic cohomology for automorphic mean curvature foliations.
J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{\mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M appears, to some extent, imposing restrictions on the mean curvature vector of the embedding.
We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold to an asymptotically Euclidean solution of the constraints on R^n. Fo…
We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat…
New restrictions found on 4-manifolds with pinched curvature.
problem Restrictions on Euler characteristic and signature of 4-manifolds with pinched curvature.
method Proved new restrictions on Euler characteristic and signature of oriented 4-manifolds with pinched sectional curvature.
result Simply connected 4-manifolds with δ≤sec≤1 are homeomorphic to S4 or CP2. Proves Besse's conjecture with a weaker condition.
problem Proving Besse's conjecture about critical metrics.
method Using a weaker condition than harmonic curvature.
result Proves Besse's conjecture for n≥3. Paper proves rigidity for certain spacelike hypersurfaces in de Sitter space.
problem Proving rigidity for hypersurfaces with curvature restrictions.
method Analogue to Guan and Shen's theorem for Riemannian space forms.
result Rigidity theorem for locally isometric hypersurfaces in de Sitter space.
Paper proves conjecture about critical metrics with divergence-free Bach tensor.
problem Proving conjecture about critical metrics with specific curvature properties.
method Used divergence-free Bach tensor to prove conjecture.
result Proved conjecture about critical metrics with divergence-free Bach tensor.
Logarithmic improvements in eigenfunction restriction estimates on hyperbolic manifolds.
problem Logarithmic improvements of L2 geodesic restriction estimates for eigenfunctions. method Adapting Chen and Sogge's approach, using explicit wave kernel formula, and detailed oscillatory integral estimates.
result Logarithmic improvement to (logλ)−21 in L2-restriction bounds. The Gauss-Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension 2k, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals…
In homogenous space Sol we study compact surfaces with constant mean curvature and with non-empty boundary. We ask how the geometry of the boundary curve imposes restrictions over all possible configurations that the surface can adopt. We obtain a flux formula and we establish results that assert that, under some restr…
Comparisons on L2n-norms of scalar curvatures between Riemannian metrics and standard metrics are obtained. The metrics are restricted to conformal classes or under certain curvature conditions.
The paper proves gap properties for critical metrics under specific conditions.
problem Proving gap properties for critical metrics under divergence-free Bach tensor condition.
method Analyzing critical point equation of total scalar curvature with divergence-free Bach tensor.
result Proves gap properties for n≥5 and a similar condition for n=4. The paper examines four-dimensional manifolds with specific curvature constraints.
problem Investigating the properties of four-dimensional manifolds under curvature constraints.
method Examined various pinching curvature conditions and classified manifolds based on these conditions.
result Closed four-dimensional manifolds under certain curvature constraints are definite or self-dual.
The study restricts scalar curvature on certain noncompact manifolds.
problem Ruling out complete metrics with nonnegative scalar curvature.
method Conditions and examples of positive scalar curvature metrics.
result Examples of manifolds with positive scalar curvature.
Develops structure theory for Ricci shrinkers without curvature restrictions.
problem Understanding the structure of Ricci shrinkers without curvature conditions.
method Structure theory development for non-collapsed Ricci shrinkers.
result Curvature estimates of Ricci shrinkers based on non-collapsing constant.
Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.
problem Rigidity of ancient solutions to mean curvature flow with convex Gauss image.
method Refined curvature estimates.
result Better rigidity theorems for ancient solutions in higher codimension.
We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a TT∗ argument, simply by using the L2-boundedness of the Hilbert transform on R, we are able to improve the corresponding L2-restriction bounds of Burq, Gérard …
The study explores Kähler manifolds with positive orthogonal Ricci curvature and finds restrictions on their geometry.
problem Understanding Kähler manifolds with positive orthogonal Ricci curvature.
method Examples and geometric consequences of the condition Ric⊥>0. result Classification results in dimensions three and four.
The study examines geometric properties of complex Hermitian manifolds and their holonomy groups.
problem Understanding the geometric properties and restrictions of Hermitian manifolds and their holonomy groups.
method Analyzing the representation of restricted holonomy groups and their geometric consequences.
result Established criteria for when a Hermitian manifold is Kähler or projective based on its holonomy group.
Ancient Lagrangian flows get limited convex solutions.
problem Controlling convex solutions of Lagrangian flows at antiquity.
method Proving a Liouville type theorem with quadratic growth restrictions.
result Ancient convex solutions are unique.
In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature bounded from below, it shows that such a manifold is of finite topological type under some restrictions shown below.
New metrics found without topological restrictions.
problem Finding metrics with constant negative scalar-Weyl curvature.
method Extended Aubin's construction to prove existence.
result Every manifold admits a metric with constant negative scalar-Weyl curvature.
Properties of general Legendrian cycles T acting in Rd×Sd−1 are studied. In particular, we give short proofs for certain uniqueness theorems with respect to the projections on the first and second component of such currents: In general, T is determined by its restriction to the Gauss curvature…
The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-We…
Curvature estimates prove existence of smooth hypersurfaces in hyperbolic space.
problem Existence of smooth complete hypersurfaces with constant curvature in hyperbolic space.
method Deriving curvature estimates to prove existence for all curvature values.
result Existence of smooth hypersurfaces for all possible curvature values.
In this paper we prove that any Riemannian surface, with no restriction of curvature at all, can be decomposed into blocks belonging just to some of these types: generalized Y-pieces, generalized funnels and halfplanes.
Totally geodesic submanifolds in spheres have restricted curvature properties.
problem Characterizing submanifolds in spheres based on curvature conditions.
method Analyzing normal curvature, scalar curvature, and second fundamental form conditions.
result Compact pseudo-umbilical submanifolds in spheres are totally geodesic under specific curvature conditions.
Hypersurfaces of manifolds of constant nonzero sectional curvature are classificated according their restricted homogeneous holonomy groups.
Paper proves uniqueness of specific spacetime surfaces in a lightcone.
problem Proving uniqueness of surfaces of constant spacetime mean curvature in a lightcone.
method Used a fairly generic notion of asymptotic flatness to prove uniqueness.
result Unique foliation by surfaces of constant spacetime mean curvature exists under weaker assumptions.
Classifies ancient solutions to curvature flows on the sphere.
problem Classifying ancient solutions to curvature flows on the sphere.
method Geometric techniques including maximum principle, rigidity result, and Alexandrov reflection argument.
result Any convex, quasi-ancient solution must be stationary or a family of shrinking geodesic spheres.
The paper extends Ricci flow conditions to less restrictive bounds.
problem Extending Ricci flow conditions to less restrictive negative bounds.
method Generalizing known Ricci flow invariant non-negative curvature conditions to negative bounds.
result Metrics with curvature operator eigenvalues greater than -1 can be evolved by Ricci flow for some uniform time.
Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.
problem Understanding topological constraints for stable free boundary CMC surfaces in negatively curved settings.
method Established intrinsic area-length-topology inequalities via a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator.
result Explicit topological restrictions for stable free boundary CMC surfaces, showing low genus and few boundary components.
The study of Kähler metrics on domains restricts their boundary geometry.
problem Understanding the geometry of domains with negatively pinched Kähler metrics.
method Analyzing the existence and properties of negatively pinched Kähler metrics on domains.
result The boundary of a convex domain without complex subvarieties of positive domain if it admits a complete Kähler metric with pinched negative holomorphic bisectional curvature.
On a compact n-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1984 by Besse, but has yet to be proved. In this paper, we prove th…
The paper explores gaps in curvature-related metrics and rigidity.
problem Understanding gaps in curvature-related metrics and rigidity.
method Analyzes three types of gaps: spectral, metric-rigidity, and topological-rigidity.
result Proposes open problems in the field.
The study examines minimal surfaces in Riemannian products of surfaces.
problem Exploring geometric and topological restrictions on minimal surfaces in Riemannian products of surfaces.
method Analyzes totally geodesic surfaces and minimal 2-spheres, 2-tori, and 2-spheres in Riemannian products of surfaces with constant curvature.
result Generically, a totally geodesic surface in a Riemannian product is either a slice or a product of geodesics. Minimal 2-spheres and 2-tori have specific properties under certain curvature conditions.