arXiv research

Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.

A new method for computing image curvature efficiently and accurately.

Localized curvature bounds ensure harmonic maps are constant.

We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property of the Gauss image under the mean curvature flow we prove the long time existence…

Study curvature of direct image bundles in deformations of maps.

Study translating solitons in Minkowski space with prescribed Gauss image.

The curvature regularities are well-known for providing strong priors in the continuity of edges, which have been applied to a wide range of applications in image processing and computer vision. However, these models are usually non-convex, non-smooth and highly non-linear, the first-order optimal condition of which ar…

An ODE variational calculation shows that an image principle curvature ratio factor can raise the lower bound, 2(Image Area), on energy of a harmonic map of a surface into Rn. In certain situations, including all radially symmetry harmonic maps, equality is achieved.

The paper finds convex hypersurfaces with specific curvature properties.

Study on Gauss images of specific minimal surfaces with finite curvature.

Paper finds unique solutions for curved surfaces with specific gradient.

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discre…

We shall show that $q$-semipositivity of the vector bundle $E$ over a Kähler total space $\mathcal X$ implies the Griffiths-semipositivity of the $q$-th direct image of $\mathcal O(K_{\mathcal X/B}\otimes E)$. As an application, we shall give a negative-curvature criterion for the generalized Weil-Petersson metric on t…

We study that the $n-$graphs defining by smooth map $f:\Om\subset \ir{n}\to \ir{m}, m\ge 2,$ in $\ir{m+n}$ of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le\f{C}{R^2}$$ under the dimension limitations and the Gauss image restrictions. If there is no…

Automatic segmentation of auditory ossicles from CT images using Ricci curvature.

In discrete differential geometry, it is widely believed that the discrete Gaussian curvature of a polyhedral vertex star equals the algebraic area of its Gauss image. However, no complete proof has yet been described. We present an elementary proof in which we compare, for a particular normal vector, its winding numbe…

In this article we are interested in the differential geometric properties of certain higher direct images of exterior powers of the sheaf of relative differentials twisted with a line bundle. We obtain explicit curvature formulas, especially in case where the said line bundle satisfies a natural curvature assumption. …

Paper proves direct image sheaf positivity for certain Kähler fibrations.

Solves capillary curvature problems for specific p values.

Proves curvature positivity of invariant direct images in complex geometry.

The paper studies curvature properties of direct image bundles.

The paper proves the existence of specific spacelike hypersurfaces in Minkowski space.

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds of dimension $n$ and a relatively ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. We give an explicit formula for the curvature tensor of these direct images.…

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hy…

Geometric sampling of networks using curvature measures.

The paper studies curvature properties of sheaves of twisted holomorphic forms on families of compact Kähler manifolds.

CAD detects anomalies and selects prototypes using polyhedron curvature.

Holomorphic families yield metrics with explicit curvature formulas.

The paper studies hypersurfaces with constant weighted mean curvature in Gaussian space.

Study theta functions and adiabatic curvature on Abelian varieties.

Given an effectively parameterized family $f:X\to S$ of canonically polarized manifolds, the Kähler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $K_{X/S}$. We use a global elliptic equation to show that this metric is strictly positive everywhere and give estimates. The dire…

The paper extends a formula linking surface curvature to projection invariants.

A geometric account explains why 'The Dress' is ambiguous, predicting observable signatures in image processing.

The field of multiple view geometry has seen tremendous progress in reconstruction and calibration due to methods for extracting reliable point features and key developments in projective geometry. Point features, however, are not available in certain applications and result in unstructured point cloud reconstructions.…

The paper establishes lower bounds on Yang-Mills functionals for fibrations.

GeoTop resolves topological ambiguity in diagnostic imaging using geometric-topological analysis.

We study the iterations of a class of curvature image operators $Λ_p^{\varphi}$ introduced by the author in (J. Funct. Anal. 271 (2016) 2133--2165). The fixed points of these operators are the solutions of the $L_p$ Minkowski problems with the positive continuous prescribed data $\varphi$. One of our results states tha…

Study curvature in holomorphic fibration fields.

On some specified convex supporting sets of spheres, we find a generalized longitude function whose level sets are totally geodesic. Given an arbitrary (weakly) harmonic map into spheres, the composition of the generalized longitude function and harmonic map satisfies an elliptic equation of divergence type. With the a…

Paper proves total curvature for convex hypersurfaces in equiaffine space.

We prove in this article that the local image of each conformal $Q$-curvature operator of arbitrary order on the sphere admits no scalar constraint. However, we prove that identities of Kazdan--Warner type hold for its graph.

For one-parameter degenerations of compact Kähler manifolds, we determine the asymptotic behavior of the first Chern form of the direct image of a Nakano semi-positive vector bundle twisted by the relative canonical bundle, when the direct image is equipped with the L2-metric.

We relate the Lipschitz-Killing measures of a definable set $X \subset \mathbb{R}^n$ in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds of $\mathbb{R}^n$, such results were established by Langevin and Shifrin.Then we give infinitesimal versions of these results. As a corollary, we…

Authors compute Weingarten map and curvatures for SL(n, R).

Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a relative ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. Using the explicit formula for the curvature tensor of these direct images, we prove that the d…

This paper is a sequel to \cite{Berndtsson}. In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kähler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Naka…

Let $M$ be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on $M$ forms an abelian group $P(M)$ after fixing a positive scalar curvature metric. The group $P(M)$ measures the size of the space of positive scalar curvature metr…