arXiv research

Formula proves monotonicity for anisotropic minimal hypersurfaces.

We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov-Hausdorff distance to the neare…

Study finds formulas for minimal submanifolds using Möbius transformations.

In this paper, we introduce a monotonicity formula for the mean curvature flow. We also apply this monotonicity formula to study the asymptotic behavior of eternal solutions.

The paper proves monotonicity formulas for solutions in Carnot groups, resembling well-known formulas for standard Laplacian and heat equations.

New formulas limit minimal submanifolds' area in curved spaces.

The paper investigates quantitative rigidity using Colding's monotonicity formulas for Ricci curvature.

In this paper, we study monotonicity formulas of eigenvalues and entropies along the rescaled List's extended Ricci flow. We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in [8] and Cao-Hou-Ling in [3]. Moreover, we also consider monotonicity formulas of $\mathcal{F}_k$-func…

Study on biharmonic map heat flow with monotonicity formula.

Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.

In this paper, we establish a general monotonicity formula of the following elliptic system $$ Δu_i+f_i(u_1,...,u_m)=0 \quad {\rm in} Ω, \label{0.1} $$ where $Ω\subset\subset \mathbb{R}^n$ is a bounded domain, $(f_i(u_1,...,u_m))=\nabla F(\vec{u})$, and $F(\vec{u})$ is a given smooth function of $\vec{u}=(u_1,...,u_m)$…

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. R…

Motivated and inspired by the recent work of Colding [5] and Colding-Minicozzi [6] we derive several families of monotonicity formulas for manifolds with nonnegative Bakry-Emery Ricci curvature, extending the formulas in [5, 6].

Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. Kähler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae…

In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem \ref{theo rescaled}.

We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is…

Derives formulas from Green function Hessian assumption.

New proof of Positive Mass Theorem using Green's function and monotonicity formula.

New formulas derived for scalar curvature in generalized Ricci flow.

In this article we establish a local parabolic almost monotonicity formula for two phase free boundary problems on Riemannian manifolds, which is an extension of a work of Edquist-Petrosyan.

Simon type monotonicity formulas for the Willmore functional $\int | \mathbf{H} |^2$ in the hyperbolic space $\mathbb{H}^n$ and $\mathbb{S}^n$ are obtained. The formula gives a lower bound of $\int_Σ | \mathbf{H} |^2$ where $Σ^2$ is any closed surface in $\mathbb{H}^n$.

Formula proves almost monotonicity for H-minimal surfaces in Heisenberg group.

We investigate monotonicity properties of $p$-harmonic vector bundle-valued $k$-forms by studying the energy-momentum tensor associated with such a form. As a consequence, we obtain a unified proof of the monotonicity formulæ for $p$-harmonic maps and Yang-Mills connections, proving a monotonicity formula for $p$-Yang-…

Paper finds formulas for minimizing perimeter in special spaces, proving key dimensions and existence.

The paper proves monotonicity formulas for minimal connections and their applications.

Stabilization technique applied to curve shortening flow in 3D space.

Unified view of monotonicity formulas for inverse mean curvature flow and $p$-capacitary potentials.

We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.

We derive identities for general flows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the first author for mean curvature flow and other nonlinear diffusions. Our results apply in particular to Ricci flow, where they yield a local mo…

In this paper, we extend the Hamilton's gradient estimates \cite{har93} and a monotonicity formula of entropy \cite{ni04} for heat flows from smooth Riemannian manifolds to (non-smooth) metric measure spaces with appropriate Riemannian curvature-dimension condition.

In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.

A local monotonicity formula for the Yang-Mills-Higgs flow on $G$-bundles over $\mathbb{R}^{n}$ ($n>4$) is proved. It is shown that the monotone quantity coïncides on certain self-similar solutions with that appearing in existing non-local monotonicity formulæ for the Yang-Mills and Yang-Mills-Higgs flows.

In this paper, we derive a new monotonicity formula for the plurisuhbarmonic functions on complete Kähler manifolds with nonnegative bisectional curvature. As applications we derive the sharp estimates for the dimension of the spaces of holomorphic functions (sections) with polynomial growth, which in particular, parti…

Explains geometric inequalities for minimal hypersurfaces.

The paper explores capital allocation using Euler formula with VaR and ES, revealing non-monotonicity and providing estimation methods.

Consider the following coupled elliptic system of equations \begin{equation*} \label{} (-Δ)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \ \ \mathbb{R}^n , \end{equation*} where $0<s\le 2$, $p>1$, $m\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\mathbb R^n\to \mathbb R$. The qualitative behavior of solutions of…

In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to uniqueness. Among the applications here is that level sets of Green's function on …

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set $Ω\subset \mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the $p$-capacitary potentials associated with $Ω$, for every $p$ suffici…

We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as later results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to `heat spheres' is proved for heat equation …

Study on relative entropy for hypersurfaces in hyperbolic space.

We derive a monotonicity formula and classify finite Morse index solutions (positive or sign-changing, radial or not) to the following triharmonic Lane-Emden equation: \begin{equation}\nonumber (-Δ)^3 u=|u|^{p-1}u \hbox{ in } \mathbb{R}^n, \end{equation} where $p$ is below the Joseph-Lundgren exponent. As a byproduct w…

New energy functional and fields for Yang-Mills theory, proving monotonicity and vanishing theorems.

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress-energy tensor which is naturally linked to conservation law and yield monotonicity formula via the coarea formula and comparison theorem in Riemannian geometr…

In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps …

The paper studies geometric constants under modified Ricci flows with variable parameters.

In this expository article, we discuss various monotonicity formulas for parabolic and elliptic operators and explain how the analysis of the function spaces and the geometry of the underlining spaces are intertwined. After briefly discussing some of the well-known analytical applications of monotonicity for parabolic …

Study applies Huisken formula to mean curvature flow in Ricci soliton background.

New PDE systems generalize Hawking mass monotonicity.