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Geometric Representation of Classes of Concave Functions and Duality 凹函数类的几何表示和对偶性
The Journal of Geometric Analysis Pub Date : 2024-06-13 DOI: 10.1007/s12220-024-01703-9
Grigory Ivanov, Elisabeth M. Werner
{"title":"Geometric Representation of Classes of Concave Functions and Duality","authors":"Grigory Ivanov, Elisabeth M. Werner","doi":"10.1007/s12220-024-01703-9","DOIUrl":"https://doi.org/10.1007/s12220-024-01703-9","url":null,"abstract":"<p>Using a natural representation of a 1/<i>s</i>-concave function on <span>({mathbb {R}}^d)</span> as a convex set in <span>({mathbb {R}}^{d+1},)</span> we derive a simple formula for the integral of its <i>s</i>-polar. This leads to convexity properties of the integral of the <i>s</i>-polar function with respect to the center of polarity. In particular, we prove that the reciprocal of the integral of the polar function of a log-concave function is log-concave as a function of the center of polarity. Also, we define the Santaló regions for <i>s</i>-concave and log-concave functions and generalize the Santaló inequality for them in the case the origin is not the Santaló point.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Velocity-Vorticity Geometric Constraints for the Energy Conservation of 3D Ideal Incompressible Fluids 三维理想不可压缩流体能量守恒的速度-涡度几何约束条件
The Journal of Geometric Analysis Pub Date : 2024-06-12 DOI: 10.1007/s12220-024-01704-8
Luigi C. Berselli, Rossano Sannipoli
{"title":"Velocity-Vorticity Geometric Constraints for the Energy Conservation of 3D Ideal Incompressible Fluids","authors":"Luigi C. Berselli, Rossano Sannipoli","doi":"10.1007/s12220-024-01704-8","DOIUrl":"https://doi.org/10.1007/s12220-024-01704-8","url":null,"abstract":"<p>In this paper we consider the 3D Euler equations and we first prove a criterion for energy conservation for weak solutions, where the velocity satisfies additional assumptions in fractional Sobolev spaces with respect to the space variables, balanced by proper integrability with respect to time. Next, we apply the criterion to study the energy conservation of solution of the Beltrami type, carefully applying properties of products in (fractional and possibly negative) Sobolev spaces and employing a suitable bootstrap argument.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Holonomy of Spherically Symmetric Projective Finsler Metrics of Constant Curvature 恒曲率球面对称投影芬斯勒度量的整体性
The Journal of Geometric Analysis Pub Date : 2024-06-10 DOI: 10.1007/s12220-024-01691-w
Mezrag Asma, Muzsnay Zoltan
{"title":"The Holonomy of Spherically Symmetric Projective Finsler Metrics of Constant Curvature","authors":"Mezrag Asma, Muzsnay Zoltan","doi":"10.1007/s12220-024-01691-w","DOIUrl":"https://doi.org/10.1007/s12220-024-01691-w","url":null,"abstract":"<p>In this paper, we investigate the holonomy group of <i>n</i>-dimensional projective Finsler metrics of constant curvature. We establish that in the spherically symmetric case, the holonomy group is maximal, and for a simply connected manifold it is isomorphic to <span>({mathcal {D}}i!f hspace{-3pt} f_o({mathbb {S}}^{n-1}))</span>, the connected component of the identity of the group of smooth diffeomorphism on the <span>({n-1})</span>-dimensional sphere. In particular, the holonomy group of the <i>n</i>-dimensional standard Funk metric and the Bryant–Shen metrics are maximal and isomorphic to <span>({mathcal {D}}i!f hspace{-3pt} f_o({mathbb {S}}^{n-1}))</span>. These results are the firsts describing explicitly the holonomy group of <i>n</i>-dimensional Finsler manifolds in the non-Berwaldian (that is when the canonical connection is non-linear) case.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141515286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fiberwise Symmetrizations for Variational Problems on Fibred Manifolds 纤维空间上变量问题的纤维对称性
The Journal of Geometric Analysis Pub Date : 2024-06-04 DOI: 10.1007/s12220-024-01698-3
Chanyoung Sung
{"title":"Fiberwise Symmetrizations for Variational Problems on Fibred Manifolds","authors":"Chanyoung Sung","doi":"10.1007/s12220-024-01698-3","DOIUrl":"https://doi.org/10.1007/s12220-024-01698-3","url":null,"abstract":"<p>We establish a framework for fiberwise symmetrization to find a lower bound of a Dirichlet-type energy functional in a variational problem on a fibred Riemannian manifold, and use it to prove a comparison theorem of the first eigenvalue of the Laplacian on a warped product manifold.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Systolic Inequality for 2-Complexes of Maximal Cup-Length and Systolic Area of Groups 群的最大杯长和收缩面积的 2-复数的收缩不等式
The Journal of Geometric Analysis Pub Date : 2024-06-03 DOI: 10.1007/s12220-024-01696-5
Eugenio Borghini
{"title":"A Systolic Inequality for 2-Complexes of Maximal Cup-Length and Systolic Area of Groups","authors":"Eugenio Borghini","doi":"10.1007/s12220-024-01696-5","DOIUrl":"https://doi.org/10.1007/s12220-024-01696-5","url":null,"abstract":"<p>We extend a systolic inequality of Guth for Riemannian manifolds of maximal <span>({mathbb {Z}}_2)</span> cup-length to piecewise Riemannian complexes of dimension 2. As a consequence we improve the previous best universal lower bound for the systolic area of groups for a large class of groups, including free abelian and surface groups, most of irreducible 3-manifold groups, non-free Artin groups and Coxeter groups or, more generally, groups containing an element of order 2.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Decay and Global Well-Posedness of the Free-Boundary Incompressible Euler Equations with Damping 带阻尼的自由边界不可压缩欧拉方程的衰减和全局良好拟合
The Journal of Geometric Analysis Pub Date : 2024-05-31 DOI: 10.1007/s12220-024-01694-7
Jiali Lian
{"title":"Decay and Global Well-Posedness of the Free-Boundary Incompressible Euler Equations with Damping","authors":"Jiali Lian","doi":"10.1007/s12220-024-01694-7","DOIUrl":"https://doi.org/10.1007/s12220-024-01694-7","url":null,"abstract":"<p>We consider the free boundary problem for a layer of incompressible fluid lying below the atmosphere and above a rigid bottom in the horizontally infinite setting. The fluid dynamics is governed by the incompressible Euler equations with damping and gravity, and the effect of surface tension is neglected on the upper free boundary. We prove the global well-posedness of the problem with the small initial data in both 2D and 3D. One of key ideas here is to make use of the time-weighted dissipation estimates to close the nonlinear energy estimates; in particular, this implies that the Lipschitz norm of the velocity is integrable-in-time, which is significantly different from that of viscous surface waves (Guo and Tice in Anal PDE 6(6):1429–1533, 2013; Wang in Adv Math 374:107330, 2020).</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Sharp Estimate for the Genus of Embedded Surfaces in the 3-Sphere 三球面中嵌入曲面之属的精确估算
The Journal of Geometric Analysis Pub Date : 2024-05-31 DOI: 10.1007/s12220-024-01689-4
Kwok-Kun Kwong
{"title":"A Sharp Estimate for the Genus of Embedded Surfaces in the 3-Sphere","authors":"Kwok-Kun Kwong","doi":"10.1007/s12220-024-01689-4","DOIUrl":"https://doi.org/10.1007/s12220-024-01689-4","url":null,"abstract":"<p>By refining the volume estimate of Heintze and Karcher [11], we obtain a sharp pinching estimate for the genus of a surface in <span>(mathbb S^{3})</span>, which involves an integral of the norm of its traceless second fundamental form. More specifically, we show that if <i>g</i> is the genus of a closed orientable surface <span>(Sigma )</span> in a 3-dimensional orientable Riemannian manifold <i>M</i> whose sectional curvature is bounded below by 1, then <span>(4 pi ^{2} g(Sigma ) le 2left( 2 pi ^{2}-|M|right) +int _{Sigma } f(|{mathop {A}limits ^{circ }}|))</span>, where <span>( {mathop {A}limits ^{circ }} )</span> is the traceless second fundamental form and <i>f</i> is an explicit function. As a result, the space of closed orientable embedded minimal surfaces <span>(Sigma )</span> with uniformly bounded <span>(Vert AVert _{L^3(Sigma )})</span> is compact in the <span>(C^k)</span> topology for any <span>(kge 2)</span>.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"85 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of Optimal Flat Ribbons 最佳扁平带的存在
The Journal of Geometric Analysis Pub Date : 2024-05-30 DOI: 10.1007/s12220-024-01683-w
Simon Blatt, Matteo Raffaelli
{"title":"Existence of Optimal Flat Ribbons","authors":"Simon Blatt, Matteo Raffaelli","doi":"10.1007/s12220-024-01683-w","DOIUrl":"https://doi.org/10.1007/s12220-024-01683-w","url":null,"abstract":"<p>We apply the direct method of the calculus of variations to show that any nonplanar Frenet curve in <span>({mathbb {R}}^{3})</span> can be extended to an infinitely narrow flat ribbon having <i>minimal</i> bending energy. We also show that, in general, minimizers are not free of planar points, yet such points must be isolated under the mild condition that the torsion does not vanish.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Energy Equality via a Priori Bound on the Velocity for Axisymmetric 3D Navier–Stokes Equations 通过轴对称三维纳维-斯托克斯方程的速度先验约束论能量相等
The Journal of Geometric Analysis Pub Date : 2024-05-30 DOI: 10.1007/s12220-024-01701-x
Jiaqi Yang
{"title":"On the Energy Equality via a Priori Bound on the Velocity for Axisymmetric 3D Navier–Stokes Equations","authors":"Jiaqi Yang","doi":"10.1007/s12220-024-01701-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01701-x","url":null,"abstract":"<p>In this paper, we are concerned with the energy equality for axisymmetric weak solutions of the 3D Navier–Stokes equations. The classical Shinbrot condition says that if the weak solution <i>u</i> of the Navier–Stokes equations belongs <span>(L^q(0,T;L^p(mathbb {R}^3)))</span> with <span>(frac{1}{q}+frac{1}{p}=frac{1}{2})</span> and <span>(pge 4)</span>, then <i>u</i> must satisfy the energy equality. For the axisymmetric Navier–Stokes equations, in our previous paper, we found that it is enough to impose the Shinbrot condition to <span>(tilde{u}=u^re_r+u^z e_z)</span>. The recent papers (Chiun-Chuan et al., Commun PDE 34(1–3):203–232, 2009; Koch et al., Acta Math 203(1):83–105, 2009) tell us if </p><span>$$begin{aligned} |tilde{u}|le frac{1}{r},,quad 0&lt; rle 1,, end{aligned}$$</span>(0.1)<p>then <i>u</i> is smooth , therefore the energy equality holds. It is natural to ask the relation between a priori bound on the velocity and the energy conservation. The aim of this paper is to investigate this problem. We shall prove that if </p><span>$$begin{aligned} |tilde{u}|le frac{1}{r^d},,quad 0&lt; rle 1,,quad d&gt;1,, end{aligned}$$</span>(0.2)<p>and </p><span>$$begin{aligned} nabla tilde{u}in L^{frac{6d-4}{2d-1}}(0,T;L^{2}(mathbb {R}^3)),, end{aligned}$$</span>(0.3)<p>then the energy equality holds.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Harmonic and Monogenic Functions on Toroidal Domains 环形域上的谐函数和单原函数
The Journal of Geometric Analysis Pub Date : 2024-05-30 DOI: 10.1007/s12220-024-01692-9
Z. Ashtab, J. Morais, R. Michael Porter
{"title":"Harmonic and Monogenic Functions on Toroidal Domains","authors":"Z. Ashtab, J. Morais, R. Michael Porter","doi":"10.1007/s12220-024-01692-9","DOIUrl":"https://doi.org/10.1007/s12220-024-01692-9","url":null,"abstract":"<p>A standard technique for producing monogenic functions is to apply the adjoint quaternionic Fueter operator to harmonic functions. We will show that this technique does not give a complete system in <span>(L^2)</span> of a solid torus, where toroidal harmonics appear in a natural way. One reason is that this index-increasing operator fails to produce monogenic functions with zero index. Another reason is that the non-trivial topology of the torus requires taking into account a cohomology coefficient associated with monogenic functions, apparently not previously identified because it vanishes for simply connected domains. In this paper, we build a reverse-Appell basis of harmonic functions on the torus expressed in terms of classical toroidal harmonics. This means that the partial derivative of any element of the basis with respect to the axial variable is a constant multiple of another basis element with subindex increased by one. This special basis is used to construct respective bases in the real <span>(L^2)</span>-Hilbert spaces of reduced quaternion and quaternion-valued monogenic functions on toroidal domains.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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