arXiv research

Projective geometry aids in analyzing fields near compact manifolds.

The study restricts stable minimal immersions in product spaces to specific configurations.

Study on Gauduchon manifolds finds metrics for projectively flat bundles.

Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.

Study of affine and projective structures on foliated complex manifolds.

Compact Kähler manifolds with positive curvature are projective and rationally connected.

Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.

Compact Kähler manifold minus a divisor is projective space.

The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.

We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …

Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.

In this paper we define strongly projectively flatness of holomorphic maps into the complex Grassmannian manifold, which is a kind of generalization of holomorphic maps into the complex projective space and prove a rigidity of equivariant strongly projectively flat maps of compact simply connected homogeneous Kähler ma…

The study classifies holomorphic projective connections on complex threefolds.

Extended Einstein manifolds reveal new symmetries.

We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that…

Study of a 32D Rosenfeld projective plane, a symmetric space.

We conjecture that the automorphism group of a topological parallelism on real projective 3-space is compact. We prove that at least the identity component of this group is, indeed, compact.

Solves supercritical dHYM on projective manifolds with specific conditions.

In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…

Let $\overline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$. Consider an affine connection $\nabla$ on $M$ for which the boundary is at infinity. Then $\nabla$ is projectively compact of order $α$ if the projective structure defined by $\nabla$ smoothly extends to all of $\overline{M}$ in a spec…

Paper proves structure of compact Kähler 3-folds with specific bundles.

The paper proves conditions for compact Kähler manifolds to be projective or rationally connected.

We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.

The paper studies stability of CR structures on compact manifolds.

It is shown that moduli spaces of complete families of compact complex hypersurfaces in complex manifolds often come equipped canonically with projective structures satisfying some natural integrability conditions.

A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a foliation F which is both transversal to the fibers and the section s. From thi…

Compact foliations preserve entropy if leaves are strictly convex projective.

Study compact Kähler manifolds with pseudo-effective tangent bundles.

The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.

For complete affine manifolds we introduce a definition of compactification based on the projective differential geometry (i.e.\ geodesic path data) of the given connection. The definition of projective compactness involves a real parameter $α$ called the order of projective compactness. For volume preserving connectio…

We prove that the automorphism group of a topological parallelism on real projective 3-space is compact. In a preceding article it was proved that at least the connected component of the identity is compact. The present proof does not depend on that earlier result.

The paper proves convexity results for a specific type of Lie groups.

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures form an …

New criterion found for Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.

By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real p…

The paper studies the non-discrete automorphisms of projective manifolds.

The projection of a compact oriented submanifold M^{n-1} in R^{n+1} on a hyperplane P^{n} can fail to bound any region in P. We call this ``projecting to zero.'' Example: The equatorial S^1 in S^2 projects to zero in any plane containing the x_3-axis. Using currents to make this precise, we show: A lipschitz (homology)…

Minimal orbits of semi-simple Lie groups are studied and related to invariant subspaces.

3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.

The study classifies stable submanifolds in product spaces of projective spaces.

The paper proves conditions for projectivity and rational connectedness of complex manifolds with quasi-positive mixed curvature.

Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…

Let M be a compact irreducible Hermitian symmetric space and write M=G/K, with G the group of holomorphic isometries of M and K the stability group of the point of 0 in M. We determine the maximal dimension of a complex projective space embedded in M as a totally geodesic submanifold.

In this note, we prove that the holonomy map from the set of equivalence classes of projective structures of parabolic type on non compact surfaces to the set of equivalence classes of parabolic representations of the fundamental group of the surface to P SL 2 (C) is a local biholomorphism.

It is proved that for a 3-dimensional compact metrizable space X the infinite real projective space is an absolute extensor of X if and only if the real projective plane is an absolute extensor of X.

Nilmanifolds are shown to be diffeomorphic to trivial bundles over tori.

In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well-established, yet competing, notions of geographic compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper score, the convex hull score, and the Reock score, and t…

We prove that for every compact, connected, differentiable 3--manifold $M$ there is a compact complex manifold $X$ which can be obtained from projective 3--space by a sequence of smooth, real blow ups and downs such that $M$ is diffeomorphic to the set of real points of $X$. By earlier results, such an $X$ can almost n…