The paper defines new types of positivity and proves properties of Schur forms for vector bundles.
problem Defining and characterizing new types of positivity for vector bundles.
method Introducing and characterizing two types of strongly decomposable positivity, proving properties of Schur forms.
result Schur forms of strongly decomposable positive vector bundles are positive or weakly positive, answering a question of Griffiths.
The paper explores criteria for positivity of forms and proves their strong positivity in specific cases.
problem Understanding positivity of exterior forms on complex vector spaces.
method Dimensionality reduction and criteria based on Hermitian matrices.
result Strong positivity of certain forms proven by duality.
The paper proves positivity of third Chern form for certain vector bundles.
problem Proving positivity of third Chern form for Griffiths positive vector bundles.
method Analyzing mixed discriminants and Schur forms.
result Positivity of third Chern form for Griffiths positive vector bundles.
The paper proves positivity of characteristic forms for certain vector bundles.
problem Characterizing positivity conditions for vector bundles.
method Operator theory, pushforward identities, and differential forms.
result Schur polynomials in Chern forms of Nakano and Griffiths positive vector bundles are positive as differential forms.
Let (M,I,J,K) be a hyperkaehler manifold, dimRM=4n. We study positive, Dolbeault-closed (2p,0)-forms on (M,I). These forms are quaternionic analogues of the positive (p,p)-forms. We construct an injective homomorphism mapping Dolbeault-closed (2p,0)-forms to closed (n+p,n+p)-forms, and positive $(2p,…
Paper proves positivity of Chern-Weil forms for certain vector bundles.
problem Proving positivity of Chern-Weil forms for Griffiths semipositive vector bundles.
method Analyzing characteristic differential forms and Schur polynomials.
result Positivity of c1(E,h)∧c2(E,h)−c3(E,h) established. Proves existence of global positive LCK potential on certain manifolds.
problem Existence of global positive LCK potential on LCK manifolds.
method Analyzes L-valued pluri-Laplacian of a function (LCK potential) and uses properties of flat connections.
result Proves existence of global positive LCK potential on certain manifolds.
We characterise positive braid links with positive Seifert form via a finite number of forbidden minors. From this we deduce a one-to-one correspondence between prime positive braid links with positive Seifert form and simply laced Dynkin diagrams, as well as a simple classification of alternating positive braid knots.
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.
Characterizes real left symmetric algebras with positive definite Koszul form and related Kähler-Einstein structures.
problem Characterizing real left symmetric algebras with positive definite Koszul form.
method Analyzes the properties of left multiplication operators and symmetric bilinear forms.
result Provides a complete characterization of real left symmetric algebras with positive definite Koszul form.
Sharp curvature condition implies spherical space form structure.
problem Characterizing manifolds with specific curvature properties.
method Proving diffeomorphism and homeomorphism to spherical space forms using curvature conditions.
result Closed manifolds with $4 rac{1}{2}$-positive curvature operator of the second kind are spherical space forms.
The paper proves a theorem for generalized p-Kähler manifolds.
problem Characterization of compact generalized p-Kähler manifolds.
method Proof based on duality between closed and exact positive forms and currents.
result Complete unified proof of Characterization Theorem for compact generalized p-Kähler manifolds.
Study proves Kato inequality leads to definite 4-manifolds with positive curvature.
problem Positive sectional curvature and Kato type inequalities on 4-manifolds.
method Analyzes Kato type inequalities on compact Riemannian 4-manifolds with positive sectional curvature.
result Proves compact Riemannian 4-manifolds satisfying a Kato type inequality are definite.
The study classifies quasi-minimal surfaces in 4D pseudo-Riemannian space-forms with positive nullity.
problem Characterizing surfaces in pseudo-Riemannian space forms with positive nullity.
method Analyzing the relative null space and classifying quasi-minimal surfaces.
result Classifications of quasi-minimal surfaces with positive relative nullity.
The study proves umbilical points have positive measure in 3D space forms.
problem Characterizing umbilical points in 3D space forms.
method Analyzing mean curvature and proving measure properties.
result Umbilical points have positive measure in 3D space forms.
Surveying problems with positive curvature forms, focusing on Ricci flow.
problem Problems with Riemannian manifolds and positive curvature forms.
method Explains how recent Ricci flow theory can solve these problems.
result Ricci flow is well-suited for solving problems involving positive curvature.
Study of torsion forms for positive line bundles.
problem Analyse of torsion forms for line bundles.
method Investigation of asymptotic behaviour of equivariant holomorphic torsion forms.
result Equivariant extension of Puchol's result.
Study on m-positivity in Kähler manifolds with new Monge-Ampère-type equation.
problem Exploring m-positivity in Kähler manifolds and its geometric applications. method Generalizing pseudo-effective and big Bott-Chern cohomology classes, proposing a new Monge-Ampère-type equation.
result Proof of a form of uniqueness for solutions of the Monge-Ampère-type equation.
New characterizations of curvature operators for specific forms via L2-estimates.
problem Characterizing semi-positive and semi-negative curvature operators for (n,q) and (p,n)-forms. method Using L2-estimates to characterize curvature operators for (n,q) and (p,n)-forms. result New characterizations of Nakano semi-positivity and semi-negativity.
Proves a formula for push-forward of polynomial Chern forms in universal vector bundles.
problem Positivity of characteristic forms in vector bundles.
method Explicit computation of Chern curvature and use of flag bundles.
result Positivity of polynomials in Chern forms for Griffiths semipositive bundles.
We give an explicit geometric argument that Artin's braid group Bn is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical form. The left-consistent form of a braid which is positive (respectively negat…
Positive representations of surface groups in PO(p,q) form connected components of character varieties.
problem Characterizing representations of surface groups in special orthogonal groups PO(p,q).
method Using Anosov representations and root versus weight collar lemmas.
result Connected components of character varieties are formed by Θ-positive Anosov representations. New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
The study classifies CR solitons based on C0-positivity and negativity.
problem Classifying CR solitons based on curvature positivity.
method Analyzing C0-positive pseudohermitian curvature and CR torsion flow. result Closed three-dimensional CR torsion solitons are the standard Sasakian space form.
S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its pert…
Study geodesic equation on mixed-volume forms on balanced manifolds, proving existence of solutions.
problem Existence of solutions to the Calabi-Yau equation for balanced metrics.
method Introduced a L2 metric space of mixed-volume forms and derived a geodesic equation. result Existence of solutions to the Calabi-Yau equation on all balanced manifolds.
Solves Calabi-Yau equation on complex manifolds with non-positive astheno-Ricci curvature.
problem Solving the form type Calabi-Yau equation on complex manifolds.
method Defined astheno-Ricci curvature and proved existence of solution under non-positive curvature condition.
result Existence of solution for Calabi-Yau equation with non-positive astheno-Ricci curvature.
Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.
problem Non-positivity of Hirzebruch form on stable weights
method Kempf--Ness and frame-potential inequality
result Zero locus of Hirzebruch form on stable weights corresponds to flat logarithmic connections
In this paper we study the heat equation (of Hodge-Laplacian) deformation of (p,p)-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a (p,p)-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Ya…
Positive representations on surfaces have positive cross-ratios and satisfy a collar lemma.
problem Characterizing representations of surface groups with positive properties.
method Proving a collar lemma and showing positivity of cross-ratios for Θ-positive representations. result Closed subsets of representation varieties are characterized by Θ-positive representations. We relate the positivity of the curvature term in the Weitzenbock formula for the Laplacian on p-forms on a complete manifold to the existence of bounded and L2 harmonic forms. In the case where the manifold is the universal cover of a compact manifold, we obtain topological and geometric information about the compa…
Proves Riemannian positive mass theorem with singularities.
problem Proves Riemannian positive mass theorem for specific types of singular manifolds.
method Uses initial data sets with a second fundamental form to transfer convexity between different singularity components.
result Proves the theorem for manifolds with some mean-concave components and others mean-convex.
New positivity condition for Hermitian manifold curvature.
problem Generalizing curvature positivity to non-Kähler manifolds.
method Introducing a new positivity condition and deriving a Bochner formula.
result Positivity condition on certain generalized Hopf and Vaisman manifolds.
Positive curvature manifolds from smaller ones using division algebras and geodesic flow.
problem Understanding positive curvature manifolds and their properties.
method Using Conner-Kobayashi reductions and division algebras to build up positive curvature manifolds.
result Existence of two linearly independent harmonic k-forms on positive curvature manifolds.
The study connects curvature operators' positivity to manifold topology.
problem Positivity of curvature operators and their geometric implications.
method Analysis of Garding cones and positivity properties of curvature operators.
result Shifted cone conditions on curvature operators constrain manifold topology.
Proves Gerber statistic is always non-negative.
problem Verifying the positive semi-definiteness of Gerber statistic.
method Analytical proof of both forms of Gerber statistic.
result Gerber statistic is positive semi-definite.
We show that the Chebyshev polynomials form a basic block of any positive basis of the Kauffman bracket skein algebras of surfaces.
Analyzes L2-harmonic forms on curved manifolds, proving integrability conditions.
problem Analyzing integrability of L2-harmonic forms on curved manifolds. method Established L∞-estimate via Moser iteration, proved vanishing of integrable forms. result Proves that L2-harmonic forms on non-positively curved manifolds are integrable if and only if they vanish. We confirm a conjecture of Hamilton: On compact manifolds the normalized Ricci flow evolves metrics with positive curvature operators to limit metrics with constant curvature.
We prove the smoothness of the L^2-analytic torsion form on some fiber bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev space, and proving that the Novikov-Shubin invariant remains positive in the Sobolev setti…
Geodesic spheres are the only quasicomplete surfaces in 3-space-forms.
problem Classifying quasicomplete surfaces in 3-space-forms.
method Using quasicompleteness as a weaker form of completeness, the global geometry of surfaces is determined.
result Geodesic spheres are the only quasicomplete surfaces of constant extrinsic curvature in 3-space-forms.
Positive braids with at least two twists form hyperbolic knots.
problem Classifying knots formed by specific braids.
method Conditions on positive braids with at least two full twists.
result Closure of such braids forms hyperbolic knots.
The paper shows that certain 4D manifolds with specific harmonic forms are diffeomorphic to CP2.
problem Characterizing 4D Riemannian manifolds with harmonic 2-forms of constant length.
method Analyzing properties of manifolds with positive sectional curvature and harmonic 2-forms.
result Compact 4D manifolds with harmonic 2-forms of constant length are diffeomorphic to CP2 under certain conditions.
In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via th…
The complex of "stable forms" on supermanifolds is studied. Stable forms on M are represented by certain Lagrangians of "copaths" (formal systems of equations, which may or may not specify actual surfaces) on M×RD. Changes of D give rise to stability isomorphisms. The Cartan--de Rham complex made of…
The paper proves a pointwise Gysin formula for vector bundles and applies it to show positivity of polynomials.
problem Positivity of polynomials in Chern forms of Griffiths semi-positive vector bundles.
method Develops a pointwise Gysin formula for hermitian vector bundles and applies it to show positivity.
result Positivity of several polynomials in Chern forms of Griffiths semi-positive vector bundles.
We introduce the notion of a positive opetope and positive opetopic cardinals as certain finite combinatorial structures. The positive opetopic cardinals to positive-to-one polygraphs are like simple graphs to free omega-categories over omega-graphs, c.f. [MZ]. In particular, they allow us to give an explicit combinato…
The paper finds a geometric lower bound for the first positive eigenvalue of the rough Laplacian on 1-forms.
problem Estimating the first positive eigenvalue of the rough Laplacian on 1-forms.
method Establishes a geometric lower bound using assumptions on Ricci curvature, diameter, and Riemann curvature tensor.
result The first positive eigenvalue of the rough Laplacian on 1-forms is bounded below by a positive constant.