Euclidean Sphere Research Articles

Unique continuation property for biharmonic hypersurfaces in spheres

We prove a unique continuation theorem for non-minimal biharmonic hypersurfaces of spheres, based on Aronszajn’s 1957 article. Under the right hypotheses, this result shows that, for these immersions, CMC on an open subset implies globally CMC. We then deduce new rigidity theorems to support the Conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.

Null Homology Groups and Stable Currents in Warped Product Submanifolds of Euclidean Spaces

In this paper, we prove that, for compact warped product submanifolds Mn in an Euclidean space En+k, there are no stable p-currents, homology groups are vanishing, and M3 is homotopic to the Euclidean sphere S3 under various extrinsic restrictions, involving the eigenvalue of the warped function, integral Ricci curvature, and the Hessian tensor. The results in this paper can be considered an extension of Xin’s work in the framework of a compact warped product submanifold, when the base manifold is minimal in ambient manifolds.

ON Free Boundary Minimal Hypersurfaces in the Riemannian Schwarzschild Space

In contrast with the three-dimensional case (cf. Montezuma in Bull Braz Math Soc), where rotationally symmetric totally geodesic free boundary minimal surfaces have Morse index one; we prove in this work that the Morse index of a free boundary rotationally symmetric totally geodesic hypersurface of the n-dimensional Riemannnian Schwarzschild space with respect to variations that are tangential along the horizon is zero, for \(n\ge 4\). Moreover, we show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, \(n\ge 8\), with Morse index equal to 0. In addition, it is shown that, for \(n\ge 4\), there exist infinitely many non-compact free boundary minimal hypersurfaces, which are not congruent to each other, with infinite Morse index. We also study the density at infinity of a free boundary minimal hypersurface with respect to a minimal cone constructed over a minimal hypersurface of the unit Euclidean sphere. We obtain a lower bound for the density in terms of the area of the boundary of the hypersurface and the area of the minimal hypersurface in the unit sphere. This lower bound is optimal in the sense that only minimal cones achieve it.

Best $$L^{2}$$-Extension of Algebraic Polynomials from the Unit Euclidean Sphere to a Concentric Sphere

Best $$L^{2}$$-Extension of Algebraic Polynomials from the Unit Euclidean Sphere to a Concentric Sphere

On the heat content functional and its critical domains

We study and classify smooth bounded domains in an analytic Riemannian manifold which are critical for the heat content at all times t>0. We do that by first computing the first variation of the heat content, and then showing that Omega is critical if and only if it has the so-called constant flow property, so that we can use a previous classification result established in [33] and [34]. The outcome is that Omega is critical for the heat content at time t, for all t>0, if and only if Omega admits an isoparametric foliation, that is, a foliation whose leaves are all parallel to the boundary and have constant mean curvature. Then, we consider the sequence of functionals given by the exit-time moments T_1(Omega ),T_2(Omega ),dots , which generalize the torsional rigidity T_1. We prove that Omega is critical for all T_k if and only if Omega is critical for the heat content at every time t, and then we get a classification as well. The main purpose of the paper is to understand the variational properties of general isoparametric foliations and their role in PDE’s theory; in some respects they generalize the properties of the foliation of mathbf{R}^{n} by Euclidean spheres.

Stability and the index of biharmonic hypersurfaces in a Riemannian manifold

In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riemannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the normal stability index of the known biharmonic hypersurfaces in a Euclidean sphere and to prove the nonexistence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chen’s conjecture on biharmonic submanifolds.

Compact gradient $$\rho$$-Einstein soliton is isometric to the Euclidean sphere

In this paper, we have investigated some aspects of gradient \(\rho\)-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient \(\rho\)-Einstein soliton satisfying some curvature conditions is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have shown that in a non-compact gradient \(\rho\)-Einstein soliton satisfying an integral condition, the scalar curvature vanishes.

A variational restriction theorem

We establish variational estimates related to the problem of restricting the Fourier transform of a three-dimensional function to the two-dimensional Euclidean sphere. At the same time, we give a short survey of the recent field of maximal Fourier restriction theory.

Some characterizations of ρ-Einstein solitons

In this article we have showed that a gradient ρ-Einstein soliton with a vector field of bounded norm and satisfying some other conditions is isometric to the Euclidean sphere. Later, we have proved that a non-trivial complete gradient ρ-Einstein soliton with finite weighted Dirichlet integral and certain restriction on Ricci curvature must be of constant scalar curvature and steady Ricci flat. Finally, we have proved that a non-shrinking or non-expanding gradient traceless Ricci soliton possessing some conditions must be steady.

Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex

We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar heat kernel bounds are shown in the context of classical Jacobi expansions, on a ball and on a simplex. These results are more precise than the qualitatively sharp Gaussian estimates proved recently by several authors.

A Simons type integral inequality for closed submanifolds in the product space [formula omitted

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces Sn×R, where Sn is the unit Euclidean sphere, is proved. As an application, an integral inequality for pnmc closed submanifolds in Sn×R is obtained. By this, it is shown that the sharpness in this inequality is attained, in the totally umbilical hypersurfaces, and in a certain family of standard tori of the form S1(1−r2)×Sn−1(r) with 0<r<1.

Liouville quantum gravity and the Brownian map III: the conformal structure is determined

Previous works in this series have shown that an instance of a sqrt{8/3}-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the sqrt{8/3}-LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and sqrt{8/3}-LQG surfaces with other topologies.

An evolutionary design of weighted minimum networks for four points in the three-dimensional Euclidean space

Abstract We find the equations of the two interior nodes (weighted Fermat–Torricelli points) with respect to the weighted Steiner problem for four points determining a tetrahedron in R 3 \mathbb{R}^{3} . Furthermore, by applying the solution with respect to the weighted Steiner problem for a boundary tetrahedron, we calculate the twist angle between the two weighted Steiner planes formed by one edge and the line defined by the two weighted Fermat–Torricelli points and a non-neighboring edge and the line defined by the two weighted Fermat–Torricelli points. By applying the plasticity principle of quadrilaterals starting from a weighted Fermat–Torricelli tree for a boundary triangle (monad) in the sense of Leibniz, established in [A. N. Zachos, A plasticity principle of convex quadrilaterals on a convex surface of bounded specific curvature, Acta Appl. Math. 129 (2014), 81–134], we develop an evolutionary scheme of a weighted network for a boundary tetrahedron in R 3 \mathbb{R}^{3} . By introducing the inverse weighted Steiner network with two interior nodes built by two different quantities of the subconscious (remaining weights) for boundary tetrahedra, we describe the evolution of a weighted network with two nodes that have inherited a subconscious. The cancellation of the dynamic plasticity of these weighted networks may be applied to the creation of evolutionary scenarios, in order to prevent the development of a quadrilateral or tetrahedral virus (a monad that has got a subconscious) and the cancerogenesis of quadrilateral cells. We continue by giving the plasticity equations for a generalized weighted minimum network with two nodes that have got a subconscious whose vertices are replaced by Euclidean spheres. This evolutionary approach may be applied to the determination of the bond strengths of molecular structures between atoms in the sense of Pauling. We obtain the analytical solutions of the weighted Fermat–Torricelli problem for the case of pairs of equal weights or one pair of equal weights. We consider as a DNA-like chain a sequence of tetrahedra whose vertices possess some symmetrical weights. By calculating the twist angles of each sequence and by applying the weighted Fermat–Torricelli tree structures with symmetrical weights or weighted Steiner tree structures, we may approximate the curve axis of a DNA-like tree chain. Finally, we construct a minimum tree, which is not a minimal Steiner tree for some boundary symmetric tetrahedra in R 3 \mathbb{R}^{3} , which has two interior nodes with equal weights having the property that the common perpendicular of some two opposite edges passes through their midpoints. We prove that the length of this minimum tree may have length less than the length of the full Steiner tree for the same boundary symmetric tetrahedra, under some angular conditions.

Evolving Compact Locally Convex Curves and Convex Hypersurfaces

A nonlocal curvature flow is investigated to evolve compact locally convex hypersurfaces in the Euclidean space $${\mathbb {E}}^{n+1}$$ . It is shown that the flow exists globally in all dimensions and deforms the evolving curve into an m-fold circle in the plane if $$n=1$$ and drives the evolving hypersurface into a Euclidean sphere if $$n>1$$ .

Bounds for discrepancies in the Hamming space

We derive bounds for the ball Lp-discrepancies in the Hamming space for 0<p<∞ and p=∞. Sharp estimates of discrepancies have been obtained for many spaces such as the Euclidean spheres and more general compact Riemannian manifolds. In the present paper, we show that the behavior of discrepancies in the Hamming space differs fundamentally because the volume of the ball in this space depends on its radius exponentially while such a dependence for the Riemannian manifolds is polynomial.

Isometry theorem of gradient Shrinking Ricci solitons

In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have shown that a four dimensional gradient shrinking Ricci soliton satisfying some conditions is isometric to S4 or RP4 or CP2. We have also deduced a condition for the shrinking Ricci soliton to be compact with quadratic volume growth.

Characterizations of Euclidean spheres

<abstract> We use the tangential component $ \psi ^{T} $ of an immersion of a compact hypersurface of the Euclidean space $ \mathbf{E}^{m+1} $ in finding two characterizations of a sphere. In first characterization, we use $ \psi ^{T} $ as a geodesic vector field (vector field with all its trajectories geodesics) and in the second characterization, we use $ \psi ^{T} $ to annihilate the de-Rham Laplace operator on the hypersurface. </abstract>

Новые границы алгебраической константы Никольского

Пусть $M_{n}=\sup_{P\in \mathcal{P}_{n}\setminus \{0\}} \frac{\max_{x\in[-1,1]}|P(x)|}{\int_{-1}^{1}|P(x)|\,dx}$ --- константа Никольского междуравномерной и интегральной нормами для алгебраических полиномов с комплекснымикоэффициентами степени не выше $n$. D. Amir и Z. Ziegler (1976) доказали, что$0.125(n+1)^{2}\le M_{n}\le 0.5(n+1)^{2}$ для $n\ge 0$. Аналогичная оценкасверху получена T.K. Ho (1976). F. Dai, D. Gorbachev и S. Tikhonov (2019--2020)уточнили этот результат, установив, что $M_{n}=Mn^{2}+o(n^{2})$ при $n\to\infty$, где $M\in (0.141,0.192)$ --- точная константа Никольского для целыхфункций экспоненциального сферического типа в пространстве$L^{1}(\mathbb{R}^{2})$ и функций экспоненциального типа в $L^{1}(\mathbb{R})$с весом $|x|$. Мы доказываем, что для произвольного $n\ge 0$ имеем $M(n+1)^{2}\le M_{n}\leM(n+2)^{2}$, где $M\in (0.1410,0.1411)$. Данное утверждение также позволяетуточнить точную константу Джексона--Никольского для полиномов на евклидовойсфере $\mathbb{S}^{2}$. Доказательство базируется на взаимосвязи алгебраическихконстант Никольского с тригонометрическими константами Бернштейна--Никольскогои наших результатах об оценках последних (2018--2019). Также мы применяемхарактеризацию экстремального алгебраического полинома, полученную D. Amir иZ. Ziegler (1976), В.В. Арестовым и М.В. Дейкаловой (2015). С помощью этойхарактеризации мы составляем тригонометрическую систему для определения нулейэкстремального полинома, которую решаем приближенно с необходимой точностью спомощью метода Ньютона.

Ограниченный оператор сдвига для (k, 1)-обобщенного преобразования Фурье

В пространствах с весом Данкля $v_k(x)$ степенного типа на $\mathbb{R}^d$, определяемым системой корней и неотрицательной функцией кратности $k$, инвариантной относительно конечной группы отражений, построен содержательный гармонический анализ. Классический анализ Фурье на евклидовом пространстве соответствует случаю $k\equiv 0$. В 2012 году Салем Бен Саид, Кобаяши и Орстед определили двупараметрическое $(k,a)$-обобщенное преобразование Фурье, действующее в пространствах с весом $|x|^{a-2}v_k(x)$, $a&gt;0$. Наиболее интересны случаи $a=2$ и $a=1$. При $a=2$ обобщенное преобразование Фурье совпадает с преобразованием Данкля и оно хорошо изучено. В случае $a=1$гармонический анализ, важный, в частности, в задачах квантовой механики, изучен пока еще не достаточно. Одним из существенных элементов гармонического анализа является ограниченный оператор сдвига, позволяющий определить свертку и структурные характеристики функций. При $a=1$ имеется оператор сдвига $\tau^yf(x)$. Его $L^p$-ограниченность недавно установлена Салемом Бен Саидом и Делеавалом, но только на радиальных функциях и при $1\le p\le 2$. В настоящей работе предложен новый оператор обобщенного сдвига $T^tf(x)$. Он получается интегрированием оператора $\tau^yf(x)$ по единичной евклидовой сфере по переменной $y'$, $|y'|=1$, $y=ty'$. Мы доказываем, что он положителен на функциях из пространства Шварца $\mathcal{S}(\mathbb{R}^d)$, для него $T^t1=1$ и он допускает представление с вероятностной мерой. Отсюда мы выводим его $L^p$-ограниченность для всех $1\le p&lt;\infty$ и ограниченность на пространстве $C_b(\mathbb{R}^d)$ непрерывных ограниченных функций.

Spectrum of the Laplacian and the Jacobi operator on rotational CMC hypersurfaces of spheres

Let $M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be a compact cmc rotational hypersurface of the $(n+1)$-dimensional Euclidean unit sphere. Denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-\Delta f-nf-|A|^2f$ the stability or Jacobi operator. In this paper we compute the spectra of their Laplace and Jacobi operators in terms of eigenvalues of second order Hill's equations. For the minimal rotational examples, we prove that the stability index --the numbers of negative eigenvalues of the Jacobi operator counted with multiplicity -- is greater than $3 n+4$ and we also prove that there are at least 2 positive eigenvalues of the Laplacian of $M$ smaller than $n$. When $H$ is not zero, we have that every non-flat CMC rotational immersion is generated by rotating a planar profile curve along a geodesic called the axis of rotation. Let $m$ be the number of points where the maximal distance from this profile curve to the origin is achieved (we assume that the coordinates of the plane containing the profile curve has been set up so that the axis of rotation goes through the origin). Let $l$ be be the wrapping number of the profile curve. We show that the number of negative eigenvalues of the operator $J$ counted with multiplicity is at least $(2l-1)n+(2m-1)$. This result was proven for the case $n=2$ by Rossman and Sultana. They called $m$ the number of bulges or the number of necks. We will slightly change the definition of $l$ to include immersed examples that contain the axis of rotation.