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Nefton Pali

Publications and source records attributed to Nefton Pali.

At least 19 recordsLinked to original sources
arXiv2025

On the distribution of charges in a conducting needle

We study the distribution of point charges in a straight conductive needle and the electric field created by them. Starting from the bead model with $n$ point charges on the needle, we show the existence and uniqueness of an equilibrium state. We also study the differential system pertaining to the system and show that the system of point charges does not converge towards an equilibrium state. In order to move from a discrete to a continuous model we increase the number of the point charges and explain the paradoxical convergence towards a uniform distribution although the charges tend to accumulate towards the ends of the needle (both at equilibrium and for the differential system of equations describing the Newtonian motion). This convergence has to be understood as that of the distribution functions of a sequence of probabilities (whether it is at equilibrium or when the charges move). We also provide visual illustrations that help understanding the studied phenomena.

math-ph↗
arXiv2023

Explicit maximal totally real embeddings

This article is the continuation of the first named author work "On maximal totally real embeddings". For real analytic compact manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent bundle, a construction of canonical maximal totally real embeddings is known from previous works by Guillemin-Stenzel, Lempert, Lempert-Sz{\"o}ke, Sz{\"o}ke and Bielawski. The construction is based on the use of Jacobi fields, which are far from being explicit. As a consequence, the form of the corresponding complex structure has been a mystery since the very beginning. A quite simple recursive expression for such complex structures has been provided in the above cited first named author work. In our series of papers we always focus on the torsion free case. In the present paper we provide a fiberwise Taylor expansion of the canonical complex structure which is expressed in terms of symmetrization of curvature monomials and a rather simple and explicit expression of the coefficients of the expansion. Our main argument applies to far more generals settings that can be useful for the study of open questions in the theory of the embeddings in consideration. In this paper we provide also evidence for some remarkable canonical vanishing of some of the integrability equations in general settings.

math.CV↗
arXiv2019

On maximal totally real embeddings

9We consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real-analytic along the fibers of the tangent bundle. This assumption is quite natural in view of a well known existence result by Bruhat and Whitney. We provide explicit integrability equations for such complex structures in terms of the fiberwise Taylor expansion. In a particular geometric case considered in the literature, we explicit much further the fiberwise Taylor expansion of the complex structure as well as the integrability equations.

math.DG↗
arXiv2018

Multiple Lie Derivatives and Forests

We obtain a complete time expansion of the pull-back operator generated by a real analytic flow of real analytic automorphisms acting on analytic tensor sections of a manifold. Our expansion is given in terms of multiple Lie derivatives. Motivated by this expansion, we provide a rather simple and explicit estimate for higher order covariant derivatives of multiple Lie derivatives acting on smooth endomorphism sections of the tangent bundle of a manifold. We assume the covariant derivative to be torsion free. The estimate is given in terms of Dyck polynomials. The proof uses a new result on the combinatorics of rooted labeled ordered forests and Dyck polynomials.

math.DG↗
arXiv2016

Concavity of Perelman's $\mathcal{W}$-functional over the space of K\"ahler potentials

In this short note we observe that the concavity of Perelman's $\mathcal{W}$-functional over a neighborhood of a K\"ahler-Ricci soliton inside the space of K\"ahler potentials is a direct consequence of author's solution of the variational stability problem for K\"ahler-Ricci solitons. Independently, we provide a rather simple proof of this fact based on some elementary formulas obtained in our previous work.

math.DG↗
arXiv2015

Chern-Ricci invariance along G-geodesics

Over a compact oriented manifold, the space of Riemannian metrics and normalised positive volume forms admits a natural pseudo-Riemannian metric $G$, which is useful for the study of Perelman's $\mathcal{W}$ functional. We show that if the initial speed of a $G$-geodesic is $G$-orthogonal to the tangent space to the orbit of the initial point, under the action of the diffeomorphism group, then this property is preserved along all points of the $G$-geodesic. We show also that this property implies preservation of the Chern-Ricci form along such $G$-geodesics, under the extra assumption of complex aniti-invariant initial metric variation and vanishing of the Nijenhuis tensor along the $G$-geodesic. This result is useful for a slice type theorem needed for the proof of the dynamical stability of the Soliton-K\"ahler-Ricci flow.

math.DG↗
arXiv2015

On complex deformations of K\"ahler-Ricci solitons

We obtain a formal obstruction, i.e. a necessary condition for the existence of polarised complex deformations of K\"ahler-Ricci solitons. This obstruction is expressed in terms of the harmonic part of the variation of the complex structure.

math.DG↗
arXiv2014

Exact Fourier inversion formula over manifolds

We show an exact (i.e. no smooth error terms) Fourier inversion type formula for differential operators over Riemannian manifolds. This provides a coordinate free approach for the theory of pseudo-differential operators.

math.AP↗
arXiv2014

Variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism

We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along Kähler structures. Our formulas show that the principal parts of the variations are quite standard complex differential operators with particular symmetry properties on the complex decomposition of the variation of the Kähler metric. We show as application that the Soliton-Kähler-Ricci flow generated by the Soliton-Ricci flow represents a complex strictly parabolic system of the complex components of the variation of the Kähler metric.

math.DG↗
arXiv2014

The Soliton-Ricci Flow with variable volume forms

We introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previous work. We still call this new flow the Soliton-Ricci flow. It corresponds to a forward Ricci type flow up to a gauge transformation generated by the gradient of the density of the volumes. The new Soliton-Ricci flow exist for all times and represents the gradient flow of Perelman's $\mathcal{W}$ functional with respect to a pseudo-Riemannian structure over the space of metrics and normalized positive volume forms. We obtain an expression of the Hessian of the $\mathcal{W}$ functional with respect to such structure. Our expression shows the elliptic nature of this operator in directions orthogonal to the orbits obtained by the action of the group of diffeomorphism. In the case the initial data is K\"ahler then the Soliton-Ricci flow preserves the K\"ahler condition and the symplectic form. The space of tamed complex structures embeds naturally to the space of metrics and normalized positive volume forms via the Chern-Ricci map. Over such space the pseudo-Riemannian structure restricts to a Riemannian one. We perform a study of the sign of the restriction of the Hessian of the $\mathcal{W}$ functional over such space. This allows us to obtain a finite dimensional reduction, and thus the solution, of the well known problem of the stability of K\"ahler-Ricci solitons.

math.DG↗
arXiv2012

The Soliton-Ricci Flow over Compact Manifolds

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by convexity results for Perelman's $\mathcal{W}$-functional over convex subsets inside adequate subspaces of Riemannian metrics. We show indeed that the Soliton-Ricci flow is generated by the gradient flow of the restriction of Perelman's $\mathcal{W}$-functional over such subspaces. Assuming long time existence of the Soliton-Ricci flow we show exponentially fast convergence to a shrinking Ricci soliton provided that the Bakry-Emery-Ricci tensor is uniformly strictly positive with respect to the evolving metric.

math.DG↗
arXiv2012

The Soliton Kähler-Ricci Flow over Fano Manifolds

We introduce a flow of Kähler structures over Fano manifolds with formal limit at infinite time a Kähler-Ricci soliton. This flow correspond to a Perelman's modified backward Kähler-Ricci type flow that we call Soliton-Kähler-Ricci flow. It can be generated by the Soliton-Ricci flow. We assume that the Soliton-Ricci flow exists for all times and the Bakry-Emery-Ricci tensor preserve a positive uniform lower bound with respect to the evolving metric. In this case we show that the corresponding Soliton-Kähler-Ricci flow converges exponentially fast to a Kähler-Ricci soliton.

math.DG↗
arXiv2012

The total second variation of Perelman's $\mathcal{W}$-functional

We show a very simple and general total second variation formula for Perelman's $\mathcal{W}$-functional at arbitrary points in the space of Riemannian metrics. Moreover we perform a study of the properties of the variations of Kähler structures. We deduce a quite simple and general total second variation formula for Perelman's $\mathcal{W}$-functional with respect to such variations. In this case the main therm in the formula depends strongly on the variation of the complex structure. We discover also convexity of Perelman's $\mathcal{W}$-functional along particular variations over points with non-negative Bakry-Emery-Ricci tensor.

math.DG↗
arXiv2009

Lecture notes on the Ein-Popa extension result

These are lecture notes on a recent remarkable preprint of Ein-Popa, which simplifies the algebraic proof of the finite generation of the canonical ring given by the team BCHM. The Ein-Popa extension result has been translated in the analytic language by Berndson-Paun and Paun. In these notes we follow the analytic language used in Berndson-Paun and Paun. The author of this manuscript does not claim any originality of the main ideas and arguments which are due to Ein-Popa, based in their turn in the ideas of Hacon-McKernan, Takayama and Siu.

math.AG↗
arXiv2009

Degenerate complex Monge-Ampère equations over compact Kähler manifolds

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Ampère equations. This type of equations is precisely what is needed in order to construct Kähler-Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf and singular Kähler-Einstein metrics over varieties of general type.

math.DG↗