arXiv research

The study finds that only round spheres shrink self-similarly under certain curvature flows.

New convex ancient solutions found for flows by high powers of curvature.

We calculate the second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator associated to high powers of a Hermitian line bundle with non-degenerate curvature, using the method of formal power series developed by Ma and Marinescu.

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature $F=H$. We prove that a certain initial…

We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the re…

We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curva…

Study finds solutions to flows by negative curvature powers.

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to …

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar curvature, then along the flow by a positive power $α$ of the mean curvature $H$, t…

Ancient flows by curvature powers in 2D have finite entropy.

We consider flows with normal velocities equal to powers strictly larger than one of the Gauss curvature. Under such flows closed strictly convex surfaces converge to points. In his work on the square of the norm of the second fundamental form, Schnürer proposes criteria for selecting quantities that are suitable for p…

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the m-th mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a …

We establish the cancellation of the first |2j-q| terms in the diagonal asymptotic expansion of the restriction to the (0, 2j)-forms of the Bergman kernel associated to the modified spin^c Dirac operator on high tensor powers of a line bundle with mixed curvature twisted by a (non necessarily holomorphic) complex vecto…

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.

New curves defined by curvature powers studied for variational properties.

In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is strictly positive and strictly less than the inverse of the dimension of the hypers…

The paper constructs hypersurfaces translating under powers of Gauss curvature.

This paper proves the existence of self-expanders for a specific curvature flow in Minkowski space.

The paper studies how convex hypersurfaces evolve under curvature flows in space forms.

We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.

In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degener…

We show non-collapsing for the evolution of nearly spherical closed convex curves in \mathbb{R}^2 under power curvature flow using two-point-methods.

New equations reveal how cylinder power in progressive lenses depends on geodesic curvature.

We study asymptotic behavior of nonparametric hypersurfaces moving by $α$ powers of Gauss curvature $α> 1/n$. Our work generalizes the results of V. Oliker [Oli91] for $α= 1$.

Classifies surfaces translating under specific curvature flows.

Conditions for torsion-free connections with specific curvature maps are derived.

The paper improves $L^2$-estimates for Dirac-Dolbeault operators on complex manifolds.

The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.

Flow adjusts curvature to avoid a fixed region, proving bounds and regularity.

New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.

The paper proves entropy power properties on Riemannian manifolds and Ricci flows.

Geometric theory explains substitutability in market outcomes based on production constraints.

Sharp lower bound found for integral varifolds' mean curvature.

We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier component…

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_κ^{n+1}$ of constant sectional curvature $κ$ evolving in direction of its normal vector, where the speed is given by a power $β(\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case…

This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $β(\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, incl…

We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solutio…

We establish the existence of the asymptotic expansion of the Bergman kernel associated to the spin-c Dirac operators acting on high tensor powers of line bundles with non-degenerate mixed curvature (negative and positive eigenvalues) by extending the paper " On the asymptotic expansion of Bergman kernel " (math.DG/040…

The paper studies volumes of direct images for high tensor powers of ample bundles.

Study shows how curved surfaces evolve smoothly to spherical shapes.

Study compares exponential and power-law kernels in modeling high-frequency trading data.

This paper aims to systematically and comprehensively initiate a foundation for using concepts from computational differential geometry as instruments for power flow computing and research. At this point we focus our discussion on the static case, with power flow equations given by quadratic functions defined on voltag…

We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth convergence of the properly rescale…

Enhances power of covariance matrix tests for high-dimensional data.

Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature

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

Study on Monge-Ampère equations with polynomial growth rates.

By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain. The curvature of the positive bundle induces a hermitian metric on the manifold. The bound is given explicitely in terms of the volume of the domain…