Study on metrics with positive scalar curvature on spherical space forms.
problem Determining metrics with positive scalar curvature on spherical space forms.
method Analysis of moduli spaces of metrics with positive scalar curvature on topological spherical space forms.
result Determination of the number of path components of moduli spaces for dimensions at least 5 and non-simply connected forms.
Study on biharmonic hypersurfaces in spheres and space forms, proving rigidity under scalar curvature condition.
problem Characterizing biharmonic hypersurfaces in space forms.
method Proved a rigidity result and established an integral formula for biharmonic hypersurfaces.
result Rigidity result under a scalar curvature condition for biharmonic hypersurfaces in space forms.
The paper classifies Randers metrics with scalar flag curvature.
problem Classifying Finsler metrics of scalar flag curvature.
method Investigating Randers metrics under the condition that β is a Killing 1-form.
result Obtained necessary conditions for Randers metrics to be of scalar flag curvature.
Study on biconservative hypersurfaces with constant scalar curvature in space forms.
problem Characterize biconservative hypersurfaces with constant scalar curvature in space forms.
method Analyzing biconservative hypersurfaces in space forms Nn+1(c), proving properties and finding specific examples. result Proves that biconservative hypersurfaces with constant scalar curvature in N4(c) have constant mean curvature, and in N5(c), they are either rotational or constant mean curvature. The paper classifies biharmonic hypersurfaces with constant scalar curvature.
problem Classifying biharmonic hypersurfaces with constant scalar curvature.
method Analyzing biharmonic hypersurfaces in space forms and spheres.
result Supports conjectures on biharmonic submanifolds and hypersurfaces.
Paper proves unique Finsler connections for scalar forms.
problem Existence and uniqueness of Finsler connections.
method Pullback approach to global Finsler geometry, study of horizontally recurrent connections.
result Existence and uniqueness of horizontally recurrent Finsler connections for scalar forms.
In this paper we will perturb the scalar curvature of compact Kahler manifolds by incorporating it with higher Chern forms, and then show that the perturbed scalar curvature has many common properties with the unperturbed scalar curvature. In particular the perturbed scalar curvature becomes a moment map, with respect …
The abstract conjectures and proves conditions for scalar-flat Kähler surfaces with specific tensor properties.
problem Conditions for scalar-flat Kähler surfaces with special tensor properties.
method Conjecture and prove in three special cases.
result The conjecture is proven in three special cases.
The paper studies scalar curvature of self-shrinkers and proves curvature bounds.
problem Proving curvature bounds for self-shrinkers.
method Analyzing scalar curvature of self-shrinkers in Euclidean space.
result Proves that the scalar curvature R of self-shrinkers is bounded by n−1. The paper examines biconservative hypersurfaces with constant curvature in space forms.
problem Characterizing biconservative hypersurfaces with specific curvature properties.
method Analyzing hypersurfaces with four distinct principal curvatures in space forms.
result Every biconservative hypersurface has constant mean and scalar curvature.
Study on contact forms with constant curvature on CR manifolds.
problem Existence of non-homothetic contact forms with constant Tanaka-Webster scalar curvature.
method Analysis of universal covers and profinite completions of CR manifolds.
result Existence of infinitely many non-homothetic contact forms on compact CR manifolds.
Study scalar curvatures in almost Hermitian geometry and derive inequalities and characterization results.
problem Characterize and study scalar curvatures in almost Hermitian manifolds.
method Explicit formulas for Hermitian scalar curvatures, inequalities of total scalar curvatures, and characterization results.
result Derive inequalities and characterization results for specific types of metrics.
We derive some elliptic differential inequalities from the Weitzenböck formulas for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some Lp and L∞ pinching result…
New Poincaré inequality for differential forms on manifolds.
problem Developing inequalities for differential forms on manifolds.
method Proving a new Poincaré-type inequality and deriving new inequalities involving mean and scalar curvatures.
result Characterized the limiting case of a new inequality involving mean and scalar curvatures of the boundary.
The paper studies scalar curvature and harmonic forms on 3-manifolds with boundaries.
problem Estimating the Thurston norm on 3-manifolds with boundaries.
method Establishing an identity relating average Euler characteristic, scalar curvature, and mean curvature.
result Characterization of the Thurston norm via scalar curvature and harmonic norm for 3-manifolds.
The paper proves geometric rigidity using harmonic twisted spinors and scalar curvature comparison.
problem Proving geometric rigidity for closed Riemannian spin manifolds with specific properties.
method Using Gromov's exact-lift two-form method and harmonic spinors to analyze scalar curvature.
result The original metric is Einstein, and the universal cover is real hyperbolic in the positive-spectrum case.
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
Paper generalizes scalar curvature theorem to weighted manifolds.
problem Generalizing scalar curvature rigidity theorem to weighted manifolds.
method Proves a refinement of Llarull's theorem for P-scalar curvature.
result Establishes a Llarull type theorem for SkimesTn−k. This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a 3−dimensional CR compact manifold locally conformally CR equivalent to the unit sphere S3 of C2. Due to Kazdan-Warner type obstructions, conditions on the function H to be realized as a We…
Researchers calculate the precise boundary operator for interacting bulk scalar fields in AdS/CFT.
problem Understanding the precise form of boundary operators dual to interacting bulk scalar fields.
method Holographic renormalization coupled with the Caffarelli/Silvestre extension theorem.
result Boundary operator dual to a bulk scalar field is an anti-local operator, the fractional Laplacian.
The scalar curvature is redefined in generalized Kahler geometry as a moment map.
problem Defining scalar curvature in generalized Kahler geometry.
method Introducing a moment map in generalized Kahler geometry to define a generalized scalar curvature.
result Infinitesimal deformations of generalized Kahler structures with constant generalized scalar curvature are finite-dimensional.
Bound on integral of scalar curvature on non-parabolic manifolds.
problem Bounding integral of scalar curvature on non-parabolic manifolds.
method Using monotonicity formulas of Colding and Minicozzi.
result Explicit bound on asymptotic weighted scaling invariant integral of scalar curvature.
Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.
problem Understanding covariant derivatives of eigenfunctions on curved spaces.
method Analyzing the Laplace-Beltrami operator on Riemannian manifolds with constant curvature.
result Covariant derivatives of eigenfunctions are scalar multiples of the functions, and these scalars are polynomials in the eigenvalue.
Study on almost Kaehler geometry of Lie groups orbits.
problem Understanding the geometry of adjoint orbits of Lie groups.
method Explicit formulas for Chern-Ricci form, scalar curvature, and Nijenhuis tensor derived from root data.
result Explicit formulas and conditions for the Chern-Ricci form and Kaehler type quotients.
Let Mn be a biharmonic hypersurface with constant scalar curvature in a space form Mn+1(c). We show that Mn has constant mean curvature if c>0 and Mn is minimal if c≤0, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and…
We study a pointwise inequality for submanifolds in real space forms involving the scalar curvature, the normal scalar curvature and the mean curvature. We translate it into an algebraic problem, allowing us to prove a slightly weaker version of it. We also prove the conjecture for certain types of submanifolds of $\ma…
This paper determines bounds on normal scalar curvature of isoparametric hypersurface focal submanifolds.
problem Classifying points with specific conditions on isoparametric hypersurface focal submanifolds.
method Analyzing the second fundamental form and scalar curvature of focal submanifolds.
result Points with Condition A achieve an upper bound of normal scalar curvature.
In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, R…
New gap theorem for hypersurfaces with constant mean curvature in space forms.
problem Finding a gap for hypersurfaces with constant mean curvature and scalar curvature.
method Analyzing the relationship between the squared length of the second fundamental form and the mean curvature.
result Proving a new gap theorem for hypersurfaces with constant mean curvature and constant scalar curvature.
4D manifolds without positive scalar curvature but products do.
problem Existence of positive scalar curvature on manifolds and products.
method Counterexamples and product constructions.
result Found 4D manifolds without positive scalar curvature but products do.
Study finds rigidity of biconservative hypersurfaces in space forms without curvature assumptions.
problem Investigating biconservative hypersurfaces in space forms without scalar curvature assumptions.
method Introduced a novel divergence-free tensor to derive results without curvature assumptions.
result Rigidity results for biconservative hypersurfaces in space forms without scalar curvature assumptions.
The paper proves conditions for submanifolds in space forms to be homeomorphic to a sphere.
problem Conditions for submanifolds in space forms to be homeomorphic to a sphere.
method Scalar curvature and Ricci curvature conditions to prove homeomorphism.
result Optimal pinching conditions for submanifolds to be homeomorphic to a sphere.
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.
Study symplectically aspherical Kähler manifolds with unique properties.
problem Existence and properties of symplectically aspherical Kähler manifolds.
method Detailed study and analysis of geometric and topological features.
result Existence of symplectically aspherical Kähler manifolds with large fundamental groups.
Study symplectic scalar curvature on supermanifolds.
problem Define and analyze symplectic scalar curvature on supermanifolds.
method Introduced two families of odd super-Fedosov structures using graded symmetric and non-symmetric connections.
result Found non-trivial odd symplectic scalar curvature for the second family.
The paper derives inequalities for Riemannian submersions and their applications.
problem Characterizing Casorati inequalities for Riemannian submersions.
method Algebraic and geometric analysis of Casorati inequalities for normalised scalar and Casorati curvatures.
result Characterization of equality cases for Casorati inequalities in Riemannian submersions.
Study explores how scalar functionals evolve under Ricci flow.
problem Understanding the evolution of functionals involving scalar quantities under Ricci flow.
method Deriving explicit expressions for the time derivative of integrals of scalar functionals under extended Ricci flow.
result Explicit expressions for the time derivative of integrals involving scalar functionals under Ricci flow.
Estimates mean curvature, scalar curvature, shape operator in warped products.
problem Estimating geometric properties in warped product spaces.
method Local and global upper estimates for curvature and shape operator.
result Results on pseudo-hyperbolic spaces and space forms.
Given a hypersurface M of null scalar curvature in the unit sphere Sn, n≥4, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in $\Rr^{n+1}$ as a normal graph over a truncated cone generated by M. Furthermore, this graph is 1-stable if t…
Study shows convergence of certain metrics to flat torus.
problem Stability of metrics on three-torus with negative scalar curvature.
method Defined metrics and used Stern's inequality to show convergence.
result Subsequence of metrics converges to flat metric.
Study of exceptional algebroids in type IIA string theory.
problem Understanding the structure of algebroids in type IIA string theory.
method Warped compactifications and analysis of local moduli space.
result Local classification of algebroids reveals new scalar deformations and twists.
I-non-degenerate spaces are spacetimes that can be characterized uniquely by their scalar curvature invariants. The ultimate goal of the current work is to construct a basis for the scalar polynomial curvature invariants in three dimensional Lorentzian spacetimes. In particular, we seek a minimal set of alg…
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…
The generalized coherent states attached to the Jacobi group realize the squeezed states. Imposing hermitian conjugacy to the generators of the Jacobi algebra, we find out the form of the weight function appearing in the scalar product. We show effectively the orthonormality of the base functions with respect to the sc…
Quantifies scalar curvature under C0 convergence, proving a refined version in all dimensions.
problem Proving a refined quantitative bound for scalar curvature under C0 convergence. method Established the refined quantitative bound in all dimensions using smoothing techniques.
result Established the refined quantitative bound for scalar curvature in all dimensions.
Paper proves a conjecture about minimal hypersurfaces in spheres.
problem Proving a conjecture about the second gap of minimal hypersurfaces with constant scalar curvature.
method Analyzing the squared norm of the second fundamental form of minimal hypersurfaces in spheres.
result Proves the Chern conjecture about the second gap of minimal hypersurfaces in spheres.
Establishes geodesic stability for Kähler metrics, proving existence of constant scalar curvature.
problem Existence of constant scalar curvature Kähler metrics.
method Exploring metric geometry of Mabuchi geodesic rays and uniform convexity properties of Kähler metrics space.
result Essentially optimal form of Donaldson's geodesic stability conjecture proved.
We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, i…