Study variational formulas for distribution geometry, finding critical metrics.
problem Analyzing the total mixed scalar curvature of a distribution.
method Developed variational formulas for extrinsic geometry, solved Euler-Lagrange equations.
result Found critical metrics related to various geometric properties.
We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\…
Paper defines new curvature measures for foliated manifolds and proves new theorems.
problem Estimating the diameter and splitting theorems for foliated manifolds.
method Introduced weighted mixed curvatures and new conditions to update estimates.
result Updated estimates of diameter and proved new splitting theorems.
We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold (Mn,g) endowed with the complementary orthogonal distributions D1 and D2. We provide conditions for symmetric (0,2)-tensors T of a simple form (defined on M) to admit metrics g~, conformal to …
We apply conformal flows of metrics restricted to the orthogonal distribution D of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of D, and for the case …
Closed Riemannian manifolds with positive mixed sectional curvature
problem Constructing closed Riemannian manifolds with positive mixed sectional curvature
method Explicit construction using totally geodesic foliations
result Positive mixed sectional curvature alone does not imply the Ferus--Adams estimate on closed manifolds
The paper studies variations of metrics on foliated manifolds and finds solutions to specific actions.
problem Variations of metrics on foliated pseudo-Riemannian manifolds.
method Developed variation formulas and applied to Einstein-Hilbert type actions.
result Found solutions like twisted products, conformal submersions, and isoparametric foliations.
The paper examines complete Yamabe solitons with finite total scalar curvature.
problem Characterizing complete Yamabe solitons with specific curvature properties.
method Analyzing steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature.
result Steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.
The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.
problem Integral and variation formulas for mixed scalar curvature in multi-product manifolds.
method Generalizes results from pseudo-Riemannian almost product manifolds to multi-product structures.
result Generalizes formulas for mixed scalar curvature in multi-product manifolds.
New method solves complex curvature equations.
problem Solving semilinear scalar curvature equations.
method Mixed convex integration method.
result New proof of scalar curvature result.
Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.
problem Characterizing minimal hypersurfaces in S^5 with certain curvature conditions.
method Analyzing hypersurfaces with constant scalar curvature and zero Gauss curvature.
result Minimal hypersurfaces in S^5 with these curvature properties are totally geodesic.
The study preserves upper bounds of total scalar curvature in conformal classes.
problem Preserving upper bounds of total scalar curvature in conformal classes.
method Analyzing Yamabe constant and scalar curvature conditions.
result The upper bound condition of total scalar curvature is preserved in a conformal class.
Totally geodesic hypersurfaces in a sphere have small total curvature.
problem Characterizing hypersurfaces with constant scalar curvature in a sphere.
method Analyzing the total curvature of locally conformally flat hypersurfaces.
result Hypersurfaces with small total curvature are totally geodesic.
Totally geodesic minimal hypersurfaces in H5 with specific curvature properties.
problem Characterizing minimal hypersurfaces in hyperbolic space with certain curvature conditions.
method Analyzing properties of minimal hypersurfaces in H5 with constant scalar curvature and zero Gauss-Kronecker curvature. result Any complete minimal hypersurface in H5 with constant scalar curvature and zero Gauss-Kronecker curvature is totally geodesic. The paper derives an integral formula for mixed scalar curvature of singular distributions.
problem Differential geometry of singular distributions on Riemannian manifolds.
method Proves divergence theorem and Codazzi equation for singular distributions.
result Derives an integral formula for mixed scalar curvature of singular distributions.
The study preserves lower bounds of total scalar curvature under specific metric convergence.
problem Preserving lower bounds of total scalar curvature on smooth manifolds.
method Used stability of Ricci flow and heat flow with Ricci flow background.
result Lower bound of weighted total scalar curvature is preserved under specified convergence conditions.
The Riemannian Bures metric on the space of (normalized) complex positive matrices is used for parameter estimation of mixed quantum states based on repeated measurements just as the Fisher information in classical statistics. It appears also in the concept of purifications of mixed states in quantum physics. Here we d…
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.
Researchers solve curvature prescription for foliated Riemann-Cartan manifolds.
problem Prescribing the mixed scalar curvature of foliated Riemann-Cartan manifolds.
method Conformal change of structure in tangent and normal directions to leaves, reduction to leafwise elliptic equation.
result Reduction to solving a leafwise elliptic equation with three stable solutions.
Study on nonspin manifolds with spin boundary, showing nonconnectedness and nontrivial fundamental group.
problem Understanding spaces of positive scalar curvature metrics on totally nonspin manifolds.
method Analysis of positive scalar curvature metrics on manifolds with spin boundary, using propagation techniques.
result Spaces of positive scalar curvature metrics are not connected and have nontrivial fundamental groups for certain dimensions.
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. Study on total mean curvature and scalar curvature on 3-manifolds.
problem Maximizing total mean curvature on compact, mean-convex 3-manifolds.
method Additivity property and variational analog of Brown-York mass.
result Finiteness of supremum of total mean curvature on 2-sphere boundaries.
We solve the modified Kazdan-Warner problem of finding metrics with prescribed scalar curvature and unit total volume.
The paper characterizes D'Atri spaces using total scalar curvature of hemispheres.
problem Characterizing D'Atri spaces using geometric properties.
method Characterization of D'Atri spaces via total scalar curvature of geodesic hemispheres.
result A 3D Riemannian manifold is a D'Atri space if and only if the total scalar curvature of tubes about geodesic segments holds.
The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a h…
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. 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.
The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.
problem Investigating the geometric properties of Riemannian manifolds influenced by average scalar curvature.
method Analyzing the conjugate radius, average area of geodesic spheres, average volume of metric balls, and total volume of closed manifolds.
result Improves the Bishop-Gromov estimate on the average volume of metric balls and proves monotone decreasing properties of certain geometric integrals.
Constructs metrics for surfaces to show uniform positive scalar curvature.
problem Proving positive scalar curvature on surfaces and their bundles.
method Constructs complete Riemannian metrics on total spaces of vector bundles.
result Shows total space of tangent bundles on non-torus surfaces admit uniform positive scalar curvature.
Study bounds total mean curvature of fill-ins with scalar curvature constraints.
problem Bounding total mean curvature of fill-ins with scalar curvature constraints.
method Combines techniques from Shi-Tam, Shi-Wang-Wei, and recent work on systolic inequality.
result Sharp constant for total mean curvature estimate when boundary metric is flat.
We consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the total mass tends to zero.
Directly applies Kazdan--Warner results to prescribe scalar curvature on bundles.
problem Prescribing scalar curvature functions on bundles.
method Direct application of Kazdan--Warner results and variational methods.
result Determines which functions are realizable as scalar curvature functions on bundles.
The paper proves uniformization for specific curvature types on manifolds.
problem Uniformizing fourth order conformal curvature on Riemannian manifolds.
method Proving existence of conformal deformations for specific curvature conditions.
result Existence of conformal deformations for positive Yamabe invariant and total Q-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 theory of monotone Riemannian metrics on the state space of a quantum system was established by Denes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant - monotone - metric can be interpreted as an average statistical uncertainty. The present paper contributes to this su…
Proves properties of 4-manifolds with scalar curvature constraints.
problem Characterizing 4-manifolds with specific scalar curvature properties.
method Analyzes locally conformally flat metrics and uses Schoen's conjecture.
result Affirmatively answers Noronha's question about 4-manifolds with scalar curvature zero.
Study relationships between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds.
problem Find a relationship between intrinsic and extrinsic invariants of Riemannian almost k-product manifolds isometrically immersed in another Riemannian manifold.
method Establish an optimal inequality involving mixed scalar curvature and square of mean curvature.
result Optimal inequality that includes mixed scalar curvature and square of mean curvature.
The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a Kähler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the n…
Study on totally real submanifolds in (LCS)n-manifolds.
problem Characterizing properties of totally real submanifolds in (LCS)n-manifolds. method Analysis of submanifolds with respect to Levi-Civita and quarter symmetric metric connections.
result Scalar curvature of C-totally real submanifolds is same for both connections.
The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in 1980's that every CPE metric must be Einstein. We prove that a 4-dimensional CPE…
Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.
problem Existence of circle bundles over large manifolds with positive scalar curvature metrics.
method Symplectic geometry techniques.
result Infinitely many examples of macroscopically large manifolds with circle bundles of positive scalar curvature.
It is a basic tenet in complex geometry that {\it negative} curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while {\it positive} curvature corresponds to the abundance of rational curves. In this spirit, we prove in this note that a projective manifo…
We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of the base is bounded below in terms of the scalar curvatures of the total space …
Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.
problem Creating metrics with negative scalar curvature.
method Constructed two types of Eguchi-Hanson metrics.
result Found metrics with negative scalar curvature.
The paper characterizes grim hyperplanes for translating solitons in mean curvature flow.
problem Characterizing grim hyperplanes for translating solitons in mean curvature flow.
method Analyzing translating solitons with nonnegative scalar curvature and mean curvature that do not change signs on each end.
result An embedded translating soliton is either a hyperplane or a grim hyperplane if it has nonnegative scalar curvature and mean curvature that do not change signs on each end.
New Kazdan-Warner problem for equivariant metrics on manifolds.
problem Equivariant scalar curvature functions on manifolds with group actions.
method Established equivariant analogue of Kazdan-Warner trichotomy.
result New class of totally G-positive pairs with positive constant scalar curvature.
Characterizes harmonic manifolds using tube properties.
problem Understanding harmonic manifolds through tube properties.
method Analyzes properties of tubes in harmonic manifolds and geodesic segments.
result Characterizes harmonic manifolds using tube properties for geodesic segments and arc lengths.
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface M in R4 with zero scalar curvature S2, nonzero Gauss-Kronecker…