arXiv research

Every harmonic map is an intrinsic bi-harmonic map as an absolute minimizer of the intrinsic bi-energy functional, therefore intrinsic bi-harmonic map and its heat flow are more geometrically natural to study, but they are also considerably more difficult analytically than the extrinsic counterparts due to the lack of …

Study fourth-order geometric flow of shape operator for co-dimension one immersions.

Study bi-harmonic flow with forcing term on smooth curves.

The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.

Both bi-harmonic map and $f$-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study $f$-bi-harmonic maps as the critical points of the $f$-bi-energy functional $\frac{1}{2}\int_M f|τ(φ)|^2dv_{g}$. This class of maps generalizes both …

In the paper, we study variation formulas for transversally harmonic maps and bi-harmonic maps, respectively. We also study the transversal Jacobi field along a map and give several relations with infinitesimal automorphisms.

} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $Δ^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero eigenvalue $Υ_1(\Om)$ admits the isoperimetric inequality of Szegö-Weinberger type: $Υ_…

We study a class of fourth-order geometric problems modelling Willmore surfaces, conformally constrained Willmore surfaces, isoperimetrically constrained Willmore surfaces, bi-harmonic surfaces in the sense of Chen, among others. We prove several local energy estimates and derive a global gap lemma.

Classifications of all biharmonic isoparametric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces are shown. Answers in case of bounded geometry to Chen's conjecture or Caddeo, Montaldo and Piu's one on biharmonic maps into a manifold of…

Proves existence of special 2-spheres in curved 3-spaces.

Quantifies uniqueness of conformal-harmonic maps on 4-manifolds.

HTE improves PINNs for high-dimensional, high-order PDEs by reducing computational cost and memory usage.

New flows introduced for symplectic geometry.

The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.

Proves uniqueness of geometric flow in various Riemannian manifolds.

Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.

Investigate scalar curvature under geometric flows

Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.

We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped pr…

The article calculates the $\mathbb{F}$-convergence rate for Ricci flows with closed and smooth tangent flows.

Paper introduces Tensor Gauge Flow Models for better data encoding.

Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.

Ancient curve shortening flows have entropy and curvature bounds equivalent.

The study disproves rotating ancient flows in 4D space.

Simplifies residual flows to make flow-based modeling more practical.

Existence of translating solutions shown for curve diffusion flow.

Modeling bone microarchitecture adaptation using geometric flows.

Variational inference relies on flexible approximate posterior distributions. Normalizing flows provide a general recipe to construct flexible variational posteriors. We introduce Sylvester normalizing flows, which can be seen as a generalization of planar flows. Sylvester normalizing flows remove the well-known single…

The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flo…

In many fields of science, high-dimensional integration is required. Numerical methods have been developed to evaluate these complex integrals. We introduce the code i-flow, a python package that performs high-dimensional numerical integration utilizing normalizing flows. Normalizing flows are machine-learned, bijectiv…

The study examines mass drop and multiplicity in mean curvature flow.

We explore the harmonic-Ricci flow---that is, Ricci flow coupled with harmonic map flow---both as it arises naturally in certain principal bundle constructions related to Ricci flow and as a geometric flow in its own right. We demonstrate that one natural geometric context for the flow is a special case of the locally …

Using the conformally invariant Cotton tensor, we define a geometric flow, the "Cotton flow", which is exclusive to three dimensions. This flow tends to evolve the initial metrics into conformally flat ones, and is somewhat orthogonal to the Yamabe flow, the latter being a flow within a conformal class. We define an en…

The paper studies mean curvature flow in a Ricci flow background with extended Ricci flow.

Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.

New derivation of Type IIA flow metrics.

Survey of geometric flows from unified string theories.

SurVAE Flows combine VAEs and flows using surjective transformations.

Study describes global existence and convergence of flows on surfaces and fibrations.

Gauge Flow Models use a learnable Gauge Field in Generative Flow Models.

By the method of discrete Morse flows, we construct an energy reducing multiple-valued function flow. The flow we get is Holder continuous with respect to the L-2 norm. We also give another way of constructing flows in some special cases, where the flow we get behaves like ordinary heat flow.

Higher Gauge Flow Models integrate higher geometry and symmetries into Generative Flow Models.

A new geometric flow $K$-flow on 3-manifolds shrinks or preserves homogeneous spheres.

New perspective on G2-structures flow from DeTurck Laplacian.

Study $G_2$-flows reducing to complex geometry flows, focusing on $G_2$-anomaly and $G_2$-Laplacian coflow.

Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.

Study on Gaussian interpolation flows for generative modeling.

The study examines singularities in flows with curvature bounds and identifies unique tangent flows.