The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the n-volumic scalar curvature. method Using the curvature-dimension condition mCD(κ,n) and smGH-convergence. result The stability of n-volumic scalar curvature ≥κ under smGH-convergence. The paper studies special Finsler spaces with Hp-scalar curvature.
problem Characterizing and investigating Finsler spaces with specific scalar curvatures.
method Intrinsic investigation and various conditions for transformations between Finsler spaces.
result Conditions for transforming Finsler spaces of scalar curvature to those of Hp-scalar curvature. This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow. We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness. We show that, in dimensions g…
The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.
problem Conditions for positive scalar curvature on manifolds with boundaries and their doubles.
method Analyzes the relationship between boundary conditions and positive scalar curvature metrics on manifolds and their doubles.
result Provides conditions for positive scalar curvature metrics on manifolds with boundaries and their doubles.
The study examines conditions for scalar curvature and Dirac operators on singular spaces.
problem Existence of scalar curvature measures and Dirac operators on singular spaces.
method Investigation of smooth manifolds with singular Riemannian metrics.
result Sufficient conditions for the existence of scalar curvature measures and Dirac operators.
Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.
problem Obstructing the existence of positive scalar curvature metrics with mean convex boundaries.
method Atiyah-Patodi-Singer index formula, deformation principle, homotopy equivalences, higher homotopy groups.
result Construction of compact manifolds with nontrivial higher homotopy groups for positive scalar curvature metrics with mean convex boundaries.
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.
Paper finds conditions for non-Einstein relative Yamabe metrics.
problem Finding relative Yamabe metrics with positive scalar curvature.
method Sufficient condition for positive constant scalar curvature metrics on manifolds with boundary.
result Examples of non-Einstein relative Yamabe metrics with positive scalar curvature.
The degree condition affects the rigidity of maps between manifolds.
problem Investigating the degree condition for scalar curvature rigidity.
method Analyzing maps between Riemannian manifolds with scalar curvature constraints.
result The degree condition is necessary for scalar curvature rigidity but not for Ricci curvature rigidity.
Study on scalar-flat Kahler 4-manifolds with a continuous symmetry.
problem Understanding scalar-flat Kahler 4-manifolds with a Killing field.
method Analysis of manifolds with a Killing field and asymptotic conditions.
result Rigidity results that restrict the behavior of scalar-flat Kahler manifolds at infinity.
Investigates conditions for non-rigidity in extremal metrics involving scalar curvature.
problem Rigidity of extremal metrics involving scalar curvature.
method Analyzes sufficient conditions for non-rigidity and provides examples.
result Provides sufficient conditions for metrics not to be rigid.
Let M be a simply-connected closed manifold of dimension ≥5 which does not admit a metric with positive scalar curvature. We give necessary conditions for M to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of M. As a consequence any simply-connected …
Study non-positive scalar curvature on hyperbolic manifolds, finding necessary and sufficient conditions.
problem Finding metrics with non-positive scalar curvature on asymptotically hyperbolic manifolds.
method Analyzing the prescribed scalar curvature problem for non-positive scalar curvature on asymptotically hyperbolic manifolds.
result Obtained a necessary and sufficient condition for the existence of solutions.
This note relaxes conditions for Kähler metrics with bounded entropy and scalar curvature.
problem Boundedness conditions for Kähler metrics with bounded entropy and scalar curvature.
method Slightly relaxes the boundedness condition on the scalar curvature.
result Apriori estimates and C3,α estimate for the potential of the Kähler metrics under relaxed conditions. Generalizes rigidity of scalar curvature for convex domains.
problem Rigidity of scalar curvature for convex domains.
method Harmonic spinors on convex domains with boundary conditions constructed by Brendle.
result Rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain.
In this paper, we consider the scalar curvature of Yamabe solitons. In particular we show that, with natural conditions and non positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that the quadratic decay at infinity of the Ricci curvature of …
The pluriclosed flow preserves Vaisman condition on compact complex surfaces.
problem Preserving Vaisman condition under pluriclosed flow.
method Pluriclosed flow on compact complex surfaces.
result Preserves Vaisman condition if and only if starting metric has constant scalar curvature.
Study finds conditions for minimal surfaces in noncompact spaces.
problem Existence of minimal surfaces in noncompact spaces with nonnegative scalar curvature.
method Conditions on a 2-sphere and its boundary.
result Existence of a closed minimal surface homologous to the boundary.
Conditions for scalar curvature on compact manifolds under conformal deformation.
problem Finding conditions for scalar curvature functions on compact manifolds.
method Analyzing sufficient and necessary conditions for scalar curvature problems within conformal classes.
result Conditions for scalar curvature functions on various compact manifolds.
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.
The study explores metrics with constant curvature on compact manifolds.
problem Finding Hermitian metrics with constant second scalar curvature on compact manifolds.
method Analyzes Yamabe-type and elliptic equations, derives geometric consequences, and proves existence under specific curvature conditions.
result Under certain curvature conditions, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, leading to the existence of Kähler-Einstein metrics.
The paper finds conditions for Yamabe solitons' metrics to be constant scalar curvature.
problem Finding conditions for Yamabe solitons' metrics to be constant scalar curvature.
method Using properties of conformal vector fields to find sufficient conditions.
result Sufficient conditions on soliton vector fields under which their metrics are of Yamabe metrics.
Study noncompact manifolds' Chern scalar curvatures, proving existence and multiplicity.
problem Prescribing Chern scalar curvatures on noncompact manifolds.
method Solving a Kazdan-Warner type equation on noncompact non-Kähler manifolds with an analytic condition.
result Established existence results and provided a new proof of multiplicity theorem.
Paper proves existence of unique constant scalar curvature Kähler metric under certain conditions.
problem Existence of constant scalar curvature Kähler metrics on polarized manifolds.
method Direct proof using microscopic stability thresholds and conditions on the limit.
result Existence of a unique constant scalar curvature Kähler metric under specific conditions.
In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.
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 solves scalar curvature problems under conformal deformation for Riemannian manifolds.
problem Solving scalar curvature problems under conformal deformation for Riemannian manifolds.
method Pointwise conformal deformation, Yamabe equation with Dirichlet boundary conditions.
result Positive, smooth solutions to the Yamabe equation with Dirichlet boundary conditions.
Study on positive scalar curvature on complex spaces with specific conditions.
problem Existence and sufficiency of positive scalar curvature metrics on stratified spaces.
method Index-theoretic conditions and cobordism analysis.
result Sufficient conditions for positive scalar curvature on certain stratified spaces.
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.
Study crystallographic groups for positive scalar curvature conditions.
problem Examining positive and negative results for Gromov-Lawson-Rosenberg Conjecture.
method Analyzing split extensions of free abelian by cyclic groups.
result Produce infinite counterexamples for the Gromov-Lawson-Rosenberg Conjecture.
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. Paper examines conditions making Riemann solitons trivial and estimates their scalar curvature.
problem Conditions making Riemann solitons trivial and scalar curvature estimates.
method Analyzes compactness and behavior at infinity of gradient fields.
result Obtains scalar curvature estimates for certain Riemann solitons.
Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
The paper studies curvature changes on manifolds with boundary.
problem Investigating conformal deformations of curvature on manifolds with boundary.
method Establishing sufficient conditions for positive scalar curvature and mean convex boundary, exploring further deformation scenarios.
result Conditions for conformal deformations to complete metrics with positive scalar curvature and mean convex boundary.
Let X --> B be a holomorphic submersion between compact Kahler manifolds of any dimension, whose fibres and base have no non-zero holomorphic vector fields and whose fibres all admit constant scalar curvature Kahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar cu…
Study gives conditions for noncompact manifolds to have positive scalar curvature.
problem Conditions for noncompact manifolds to have positive scalar curvature.
method Analyzes finite and infinite volume cases to find obstructions.
result Provides conditions for noncompact manifolds to admit metrics with positive scalar curvature.
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.
Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.
problem Finding a conformal metric with zero scalar curvature and prescribed boundary mean curvature.
method Construction of local test functions to resolve open cases and establish new solvability conditions.
result Established new solvability conditions for the problem.
Study on scalar curvature deformations in pseudohermitian manifolds.
problem Deformation of scalar curvature in pseudohermitian manifolds.
method Analogy with Riemannian manifolds, introduction of R-singular spaces, stability conditions, partial infinitesimal rigidity. result Partial infinitesimal rigidity result for scalar curvature of compact pseudohermitian manifolds.
Study on gradient h-almost Yamabe solitons with scalar curvature estimation.
problem Exploring triviality and scalar curvature estimation of gradient h-almost Yamabe solitons.
method Established sufficient conditions for triviality and scalar curvature estimation under integral inequalities involving the scalar curvature and soliton function.
result Extended and refined former works on almost and h-almost Yamabe solitons, characterizing their geometric structures.
Estimates for metrics with constant Chern scalar curvature on complex manifolds.
problem Finding metrics with constant Chern scalar curvature on complex manifolds.
method Proving a priori estimates conditional on an upper bound on entropy.
result Extending a recent result by Chen-Cheng in the Kähler setting.
The paper explores geometry and positive scalar curvature on non-compact manifolds.
problem Understanding the relationship between geometry and positive scalar curvature on non-compact manifolds.
method Analysis of volume growth, scalar curvature integral, and width in different dimensions.
result Proves minimal volume growth and integral of scalar curvature in three dimensions, and volume growth with stronger conditions in higher dimensions.
We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant …
The doubling conjecture for positive scalar curvature is proven under certain conditions.
problem Determining when a manifold with a specific boundary condition admits positive scalar curvature.
method Surgery techniques for positive scalar and mean curvature, and existence of area-minimizing hypersurfaces.
result The doubling conjecture holds true for manifolds with certain split conditions on fundamental groups.
Modified condition proves no positive scalar curvature for enlargeable manifolds.
problem Proving no positive scalar curvature for modified Λ2-enlargeable manifolds. method Replacing constant near infinity with locally constant near infinity and proving the result.
result Modified Λ2-enlargeable manifolds cannot carry a complete Riemannian metric of positive scalar curvature. Paper investigates prescribing Chern scalar curvatures on specific manifolds.
problem Prescribing Chern scalar curvatures on noncompact Hermitian manifolds with nonpositive curvatures.
method Establishes existence results and sufficient conditions for negative curvature metrics.
result Obtains sufficient conditions for the existence of a constant negative Chern scalar curvature metric.
Proves conditions for radial Kaehler metrics to be Kaehler-Einstein.
problem Characterizing radial Kaehler metrics as Kaehler-Einstein.
method Analyzes conditions for radial Kaehler metrics to be Kaehler-Einstein.
result Conditions for radial Kaehler metrics to be Kaehler-Einstein are provided.
We use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which are all satisfied if the manifold is curvature homogeneous. This leads to a comp…