This paper extends a 3D result to higher dimensions for manifolds with positive curvature.
problem Proving higher-dimensional manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the Euclidean disk.
method Using the positive curvature operator and strictly convex boundary conditions to deduce the manifold's diffeomorphism to the Euclidean disk.
result Compact n-manifolds with positive curvature operator and strictly convex boundary are diffeomorphic to the standard n-dimensional Euclidean disk.
This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is pro…
Algorithm extends Euclidean cell decomposition to projective surfaces.
problem Computing Euclidean cell decomposition for non-hyperbolic surfaces.
method Generalised Weeks' algorithm to strictly convex projective surfaces.
result Algorithm successfully decomposes Euclidean cell structure for projective surfaces.
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
Study convex projective manifolds with compact and convex ends.
problem Holonomy of convex projective manifolds.
method Extend Koszul's theorem to non-compact manifolds with boundary.
result Holonomies of properly convex structures form an open subset of the representation variety.
Compact foliations preserve entropy if leaves are strictly convex projective.
problem Entropy rigidity for foliations by strictly convex projective manifolds.
method Analysis of foliated volume entropies and homeomorphisms.
result Equality in foliated volume entropies implies homothetic leaves.
Proof that convex structures on manifolds are open and closed.
problem Characterizing the set of holonomies for convex structures.
method Expository proof using properties of holonomies and projective structures.
result The set of holonomies forms an open and closed subset of the space of representations.
The study confirms Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary.
problem Proving Liouville-type theorems for positive harmonic functions on specific types of manifolds.
method Employing the P-function method and a closed conformal vector field inherent to such manifolds.
result Confirms some cases of Wang's conjecture and provides a partial verification of Wang's conjecture on warped product manifolds.
In this article, we study convex affine domains which can cover a compact affine manifold. For this purpose, we first show that every strictly convex quasi-homogeneous projective domain has at least C1 boundary and it is an ellipsoid if its boundary is twice differentiable. And then we show that an n-dimensional par…
The paper bends hyperbolic manifolds into convex structures.
problem Creating convex structures from hyperbolic manifolds.
method Bending totally geodesic hypersurfaces in finite volume hyperbolic manifolds.
result Examples of non-compact, finite volume, properly convex manifolds are found.
Injective transform for manifolds with convex boundary, solving integral geometry problems.
problem Injectivity of the geodesic X-ray transform with matrix weights.
method Reduction to local invertibility near strict convexity, detailed layer stripping analysis.
result Connection and Higgs field uniquely determined by scattering relation.
The study proves Liouville theorems on curved manifolds with convex boundaries.
problem Proving Liouville theorems on manifolds with nonnegative curvature and strictly convex boundary.
method Analyzing smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary.
result Derives Liouville theorems and verifies a conjecture about eigenvalues and inequalities.
Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its topological entropy is less than n-1, with equality if and only if the structure is Riemannian, that is hyperbolic. As a corol…
Geodesic tomography identifies piecewise constants on convex manifolds.
problem Determining piecewise constant functions on nontrapping manifolds.
method Iterating local uniqueness results based on geodesic integrals.
result Piecewise constant functions are uniquely determined by their geodesic integrals.
Paper reconstructs compact Riemannian manifolds from travel time data.
problem Reconstructing compact Riemannian manifolds from partial travel time data.
method Embedding in function space, studying distance function regularity.
result Reconstruction of compact Riemannian manifolds from travel time data.
Study of manifolds with specific curvature properties using capillary surfaces.
problem Obtaining geometric properties of manifolds with nonnegative scalar curvature and strictly mean convex boundary.
method Use of stable capillary surfaces and Urysohn width to study geometric properties.
result Obtained an obstruction to filling 2-manifolds by 3-manifolds.
Local invertibility of ray transforms on convex manifolds.
problem Invertibility of ray transforms on compact Riemannian manifolds with strictly convex boundary.
method Local invertibility results for transverse and mixed ray transforms of 1 and 1+1 tensors.
result Local invertibility of ray transforms near boundary points, leading to global results.
We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy anyone of the following three types of (new) boundary: strictly convex, totally geode…
K-energy is not strictly convex on certain complex manifolds, but under specific conditions, it is.
problem The strict convexity of Mabuchi's K-energy on geodesically complete spaces of bounded positive forms.
method Simple toric example and further assumptions on toric manifolds.
result Strict convexity holds in the toric case under certain conditions, leading to a uniqueness result.
The renormalized volume of hyperbolic 3-manifolds is strictly convex.
problem Understanding the convexity of the renormalized volume in hyperbolic 3-manifolds.
method Analyzing the Hessian of the renormalized volume and its positivity.
result Strict convexity of the functional volume of the convex core at its minimum point.
Local invertibility of higher rank tensor fields on curved manifolds proven.
problem Local invertibility of geodesic ray transform on tensor fields of rank four.
method Proved local invertibility up to potential fields on Riemannian manifolds with strictly convex boundary.
result Local invertibility of tensor fields of rank four on curved manifolds proven.
Local invertibility of higher order tensor transforms on compact manifolds.
problem Invertibility of higher order tensor transforms on compact manifolds.
method Local invertibility of transverse and mixed ray transforms of tensors on compact Riemannian manifolds.
result Local invertibility of transverse and mixed ray transforms of tensors for specific dimensions.
We extend the canonical cell decomposition due to Epstein and Penner of a hyperbolic manifold with cusps to the strictly convex setting. It follows that a sufficiently small deformation of the holonomy of a finite volume strictly convex real projective manifold is the holonomy of some nearby projective structure with r…
Researchers develop Q-curvature for convex hypersurfaces using ambient metrics.
problem Calculating Q-curvature for convex hypersurfaces in projective manifolds.
method Using the ambient metric, they construct GJMS operators and relate Q-curvature to the logarithmic coefficient in volume expansion.
result Derived first and second variation formulas for strictly convex domains.
Study volume expansion on convex domains using Blaschke metric.
problem Volume expansion of Blaschke metric on strictly convex domains.
method Expressed logarithmic coefficient L as integrals of affine invariants over the boundary and formulated intrinsic geometry as conformal Codazzi structure.
result L is a global conformal invariant of the boundary.
3-manifolds with convex boundary are rigid in certain curvature conditions.
problem Characterizing the rigidity of nonpositively curved manifolds with convex boundaries.
method Using a comparison formula for total curvature of Riemannian hypersurfaces, the authors prove the rigidity of the manifolds.
result Compact Riemannian 3-manifolds with strictly convex simply connected boundary and sectional curvature K≤a≤0 are isometric to a convex domain in a complete simply connected space of constant curvature a.
Study on projective maps between real projective manifolds, showing finite homotopy classes and structures.
problem Understanding the structure of projective maps between real projective manifolds.
method Analyzing the set of projective maps and their homotopy classes, proving finite structures and properties.
result Each homotopy class has a structure of a real projective manifold, with constraints on non-trivial classes.
We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature − including strictly convex domains of the Euclidean space R3.
The paper explores convex functions on Riemannian manifolds and their geometric properties.
problem Existence and non-existence of convex functions on Riemannian manifolds.
method Analyzes geometric properties and conditions for the existence of convex functions on Riemannian manifolds.
result Geometric conditions ensuring the existence of convex functions on certain manifolds.
The paper proves geodesic connectedness for convex functions in space-times.
problem Geodesic connectedness of space-times and semi-Riemannian manifolds.
method Geometric-topological proofs for specific classes of space-times.
result Geodesic connectedness for null-disprisoning space-times and timelike strictly convex hypersurfaces.
Vanishing of equivariant cohomology groups for proper Lie group actions.
problem Vanishing of equivariant differentiable cohomology groups for proper Lie group actions.
method Establishing vanishing of equivariant differentiable cohomology groups with coefficients in C∞-functions. result The canonical class in the first differential cohomology of G with coefficients in C∞-functions on M vanishes if and only if G acts properly on M. In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-triv…
We consider Gromov-Thurston examples of negatively curved n-manifolds which do not admit metrics of constant sectional curvature. We show that for each n some of the Gromov-Thurston manifolds admit strictly convex real-projective structures.
Every convex set in a generic Riemannian manifold has peculiar properties.
problem Characterizing convex sets in Riemannian manifolds.
method Analyzing geodesics and hypersurfaces in Riemannian manifolds.
result Convex sets in generic Riemannian manifolds are strictly convex if bounded by smooth hypersurfaces.
The study pinches conditions for constant mean curvature surfaces in convex 3-manifolds.
problem Understanding the topology and geometry of constant mean curvature surfaces with free boundaries in convex 3-manifolds.
method Provided pinching conditions on the traceless second fundamental form to guarantee surface topology.
result The surface is either a disk, annulus, spherical cap, or Delaunay surface under certain conditions.
Sharp bounds on minimal surfaces in compact 3-manifolds with convex boundary.
problem Finding bounds on the length of minimal surfaces in compact 3-manifolds.
method Using Toponogov's theorem and geometric inequalities, the authors derive sharp upper bounds for the boundary length of minimal surfaces.
result If the boundary length of a minimal surface saturates the upper bound, the manifold must be a Euclidean 3-ball and the surface a Euclidean disk.
Sharp lower bound for curvature in Kähler manifolds.
problem Curvature bounds for real hypersurfaces in Kähler manifolds.
method Gauss equation for semi-isometric CR immersions.
result Proves $ rac12$-positivity of Tanaka-Webster scalar curvature.
Study on compact manifolds with specific curvature and boundary properties.
problem Compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary.
method New approach to study these manifolds, formulating conjectures and establishing partial results.
result Support for conjectures on these manifolds.
Lower bound found for Steklov eigenvalue on curved manifolds.
problem Finding bounds for Steklov eigenvalues on curved spaces.
method Established a new lower bound using geometric curvature conditions.
result Found a new lower bound for the first non-zero Steklov eigenvalue.
Transforms uniquely determine Higgs fields on real-analytic manifolds.
problem Determining Higgs fields from transforms on manifolds.
method Matrix-weighted real-analytic double fibration transforms.
result Higgs fields can be uniquely determined from transforms.
The paper studies quasi-X-convex functions and their applications in optimization.
problem Optimization problems with quasi-X-convex functions. method Definition and study of X-convex, quasi-X-convex, and related functions. result Applications of quasi-X-convex functions in optimization problems. Strict convexity of Wulff shapes linked to C1 integrands.
problem Characterizing strictly convex Wulff shapes.
method Analyzing convex integrands of class C1. result Wulff shapes are strictly convex if and only if their integrands are C1. A strictly convex real projective orbifold is equipped with a natural Finsler metric called the Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to …
The study shows how strictly convex domains in Euclidean spaces are rigid.
problem Understanding the rigidity of strictly convex domains in Euclidean spaces.
method Proved a rigidity theorem for smooth strictly convex domains in Euclidean spaces.
result Smooth strictly convex domains in Euclidean spaces are rigid.
Study finds strictly convex surfaces with specific curvature and boundary in space forms.
problem Finding strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms.
method Using C2 a priori estimates and degree theory arguments, the study establishes existence results. result Existence of strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms.
Proves path connectedness and contractibility of metric spaces with non-negative Ricci curvature.
problem Existence of metrics with non-negative Ricci curvature on convex three-manifolds.
method Path connectedness and contractibility proofs using moduli spaces.
result Existence of properly embedded free boundary minimal annuli.
The isoperimetric profile remains continuous for certain Riemannian manifolds.
problem Continuity of the isoperimetric profile in Riemannian manifolds.
method Analyzing complete Riemannian manifolds with convex exhaustion functions.
result The isoperimetric profile is continuous and non-decreasing.
Study local magnetic ray transform of tensor fields on Riemannian manifolds.
problem Stable inversion of magnetic ray transforms of tensor fields up to rank two.
method Analyzes magnetic ray transforms of tensor fields of different orders on Riemannian manifolds with boundary.
result Stable inversion of magnetic ray transforms near strictly convex boundary points.