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Harmonic Spinors in the Ricci Flow 利玛窦流中的谐波旋光子
The Journal of Geometric Analysis Pub Date : 2024-05-16 DOI: 10.1007/s12220-024-01665-y
Julius Baldauf
{"title":"Harmonic Spinors in the Ricci Flow","authors":"Julius Baldauf","doi":"10.1007/s12220-024-01665-y","DOIUrl":"https://doi.org/10.1007/s12220-024-01665-y","url":null,"abstract":"<p>This paper provides a new definition of the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman’s Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg–Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin–Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060589","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 Properties of Quasi-Banach Function Spaces 论准巴拿赫函数空间的性质
The Journal of Geometric Analysis Pub Date : 2024-05-14 DOI: 10.1007/s12220-024-01673-y
Aleš Nekvinda, Dalimil Peša
{"title":"On the Properties of Quasi-Banach Function Spaces","authors":"Aleš Nekvinda, Dalimil Peša","doi":"10.1007/s12220-024-01673-y","DOIUrl":"https://doi.org/10.1007/s12220-024-01673-y","url":null,"abstract":"<p>In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they possess a generalised version of Riesz–Fischer property, that embeddings between them are always continuous, and that the dilation operator is bounded on them. We also provide a characterisation of separability for quasi-Banach function spaces over the Euclidean space. Furthermore, we extend the classical Riesz–Fischer theorem to the context of quasinormed spaces and, as a consequence, obtain an alternative proof of completeness of quasi-Banach function spaces.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060456","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
Non-quadratic Euclidean Complete Affine Maximal Type Hypersurfaces for $$theta in (0,(N-1)/N]$$ $$theta in (0,(N-1)/N]$$ 的非四边形欧几里得完全仿射最大类型超曲面
The Journal of Geometric Analysis Pub Date : 2024-05-13 DOI: 10.1007/s12220-024-01678-7
Shi-Zhong Du
{"title":"Non-quadratic Euclidean Complete Affine Maximal Type Hypersurfaces for $$theta in (0,(N-1)/N]$$","authors":"Shi-Zhong Du","doi":"10.1007/s12220-024-01678-7","DOIUrl":"https://doi.org/10.1007/s12220-024-01678-7","url":null,"abstract":"<p>Bernstein problem for affine maximal type equation </p><span>$$begin{aligned} u^{ij}D_{ij}w=0, wequiv [det D^2u]^{-theta }, forall xin Omega subset {mathbb {R}}^N end{aligned}$$</span>(0.1)<p>has been a core problem in affine geometry. A conjecture (Version I in Section 1) initially proposed by Chern (Proc. Japan-United States Sem., Tokyo, 1977, 17-30) for entire graph with <span>(N=2, theta =3/4)</span> and then was strengthened by Trudinger-Wang (Invent. Math., <b>140</b>, 2000, 399-422) to its full generality (Version II), which asserts that any Euclidean complete, affine maximal, locally uniformly convex <span>(C^4)</span>-hypersurface in <span>({mathbb {R}}^{N+1})</span> must be an elliptic paraboloid. At the same time, the Chern’s conjecture was solved completely by Trudinger-Wang in dimension two. Soon after, the Affine Bernstein Conjecture (Version III) for affine complete affine maximal hypersurfaces was also shown by Trudinger-Wang in (Invent. Math., <b>150</b>, 2002, 45-60). Thereafter, the Bernstein problem has morphed into a broader conjectures for any dimension <span>(Nge 2)</span> and any positive constant <span>(theta &gt;0)</span>. The Bernstein theorem of Trudinger-Wang was then generalized by Li-Jia (Results Math., <b>56</b> 2009, 109-139) to <span>(N=2, theta in (3/4,1])</span> (see also Zhou (Calc. Var. PDEs., <b>43</b> 2012, 25-44) for a different proof). In the past twenty years, much effort was done toward higher dimensional issues but not really successful yet, even for the case of dimension <span>(N=3)</span>. Recently, counter examples were found in (J. Differential Equations, <b>269</b> (2020), 7429-7469), toward the Full Bernstein Problem IV for <span>(Nge 3,theta in (1/2,(N-1)/N))</span> and using a much more complicated argument. In this paper, we will construct explicitly various new Euclidean complete affine maximal type hypersurfaces which are not elliptic paraboloid for the improved range </p><span>$$begin{aligned} Nge 2, theta in (0,(N-1)/N]. end{aligned}$$</span>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141530802","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 Metric Lower Bound Estimate for Geodesics in the Space of Kähler Potentials 凯勒势空间中大地线的度量下限估计
The Journal of Geometric Analysis Pub Date : 2024-05-12 DOI: 10.1007/s12220-024-01654-1
Jingchen Hu
{"title":"A Metric Lower Bound Estimate for Geodesics in the Space of Kähler Potentials","authors":"Jingchen Hu","doi":"10.1007/s12220-024-01654-1","DOIUrl":"https://doi.org/10.1007/s12220-024-01654-1","url":null,"abstract":"<p>In this paper, we establish a positive lower bound estimate for the second smallest eigenvalue of the complex Hessian of solutions to a degenerate complex Monge–Ampère equation. As a consequence, we find that in the space of Kähler potentials any two points close to each other in <span>(C^2)</span> norm can be connected by a geodesic along which the associated metrics do not degenerate.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927657","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 Relaxation of Gauss’s Capillarity Theory Under Spanning Conditions 论跨度条件下高斯毛细管理论的松弛
The Journal of Geometric Analysis Pub Date : 2024-05-12 DOI: 10.1007/s12220-024-01675-w
Michael Novack
{"title":"On the Relaxation of Gauss’s Capillarity Theory Under Spanning Conditions","authors":"Michael Novack","doi":"10.1007/s12220-024-01675-w","DOIUrl":"https://doi.org/10.1007/s12220-024-01675-w","url":null,"abstract":"<p>We study a variational model for soap films in which the films are represented by sets with fixed small volume rather than surfaces. In this problem, a minimizing sequence of completely “wet\" films, or sets of finite perimeter spanning a wire frame, may converge to a film containing both wet regions of positive volume and collapsed (dry) surfaces. When collapsing occurs, these limiting objects lie outside the original minimization class and instead are admissible for a relaxed problem. Here we show that the relaxation and the original formulation are equivalent by approximating the collapsed films in the relaxed class by wet films in the original class.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927753","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
New Estimations for Coisotropic Ekeland–Hofer–Zehnder Capacity 各向同性 Ekeland-Hofer-Zehnder 容量的新估算值
The Journal of Geometric Analysis Pub Date : 2024-05-12 DOI: 10.1007/s12220-024-01672-z
Kun Shi
{"title":"New Estimations for Coisotropic Ekeland–Hofer–Zehnder Capacity","authors":"Kun Shi","doi":"10.1007/s12220-024-01672-z","DOIUrl":"https://doi.org/10.1007/s12220-024-01672-z","url":null,"abstract":"<p>In this paper, we give an estimation for coisotropic Ekeland–Hofer–Zehnder capacity by combinatorial formula. This result implies that coisotropic Ekeland–Hofer–Zehnder capacity can measure the symmetry of convex bodies with respected to <span>(mathbb {R}^{n,k})</span> in some sense. Next, we talk about the behavior of coisotropic Ekeland–Hofer–Zehnder capacity of convex domains in the classical phase space with respect to symplectic <i>p</i>-products.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927660","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
Local Well-Posedness and Regularity Criterion for the 3D Density-Dependent Incompressible Maxwell–Navier–Stokes System 依赖密度的三维不可压缩麦克斯韦-纳维尔-斯托克斯系统的局部拟合性和正则性准则
The Journal of Geometric Analysis Pub Date : 2024-05-12 DOI: 10.1007/s12220-024-01659-w
Jishan Fan, Yong Zhou
{"title":"Local Well-Posedness and Regularity Criterion for the 3D Density-Dependent Incompressible Maxwell–Navier–Stokes System","authors":"Jishan Fan, Yong Zhou","doi":"10.1007/s12220-024-01659-w","DOIUrl":"https://doi.org/10.1007/s12220-024-01659-w","url":null,"abstract":"<p>In this paper, we prove the local well-posedness and regularity criterion of strong solutions to the 3D density-dependent incompressible Maxwell–Navier–Stokes system in a bounded and simply connected domain with vacuum.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927763","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
Characterizations of Weights in Martingale Spaces 马丁格尔空间中权重的特征
The Journal of Geometric Analysis Pub Date : 2024-05-10 DOI: 10.1007/s12220-024-01674-x
Jie Ju, Wei Chen, Jingya Cui, Chao Zhang
{"title":"Characterizations of Weights in Martingale Spaces","authors":"Jie Ju, Wei Chen, Jingya Cui, Chao Zhang","doi":"10.1007/s12220-024-01674-x","DOIUrl":"https://doi.org/10.1007/s12220-024-01674-x","url":null,"abstract":"<p>Grafakos systematically proved that <span>(A_infty )</span> weights have different characterizations for cubes in Euclidean spaces in his classical text book. Very recently, Duoandikoetxea, Martín-Reyes, Ombrosi and Kosz discussed several characterizations of the <span>(A_{infty })</span> weights in the setting of general bases. By conditional expectations, we study <span>(A_infty )</span> weights in martingale spaces. Because conditional expectations are Radon–Nikodým derivatives with respect to sub<span>(hbox {-}sigma hbox {-})</span>fields which have no geometric structures, we need new ingredients. Under a regularity assumption on weights, we obtain equivalent characterizations of the <span>(A_{infty })</span> weights. Moreover, using weights modulo conditional expectations, we have one-way implications of different characterizations.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927751","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 Morse Property of Limit Functions Appearing in Mean Field Equations on Surfaces with Boundary 有边界曲面上均值场方程中出现的极限函数的莫尔斯特性
The Journal of Geometric Analysis Pub Date : 2024-05-09 DOI: 10.1007/s12220-024-01664-z
Zhengni Hu, Thomas Bartsch
{"title":"The Morse Property of Limit Functions Appearing in Mean Field Equations on Surfaces with Boundary","authors":"Zhengni Hu, Thomas Bartsch","doi":"10.1007/s12220-024-01664-z","DOIUrl":"https://doi.org/10.1007/s12220-024-01664-z","url":null,"abstract":"<p>In this paper, we study the Morse property for functions related to limit functions of mean field equations on a smooth, compact surface <span>(Sigma )</span> with boundary <span>(partial Sigma )</span>. Given a Riemannian metric <i>g</i> on <span>(Sigma )</span> we consider functions of the form\u0000</p><p>where <span>(sigma _i ne 0)</span> for <span>(i=1,ldots ,m)</span>, <span>(G^g)</span> is the Green function of the Laplace-Beltrami operator on <span>((Sigma ,g))</span> with Neumann boundary conditions, <span>(R^g)</span> is the corresponding Robin function, and <span>(h in {{mathcal {C}}}^{2}(Sigma ^m,mathbb {R}))</span> is arbitrary. We prove that for any Riemannian metric <i>g</i>, there exists a metric <span>(widetilde{g})</span> which is arbitrarily close to <i>g</i> and in the conformal class of <i>g</i> such that <span>(f_{widetilde{g}})</span> is a Morse function. Furthermore we show that, if all <span>(sigma _i&gt;0)</span>, then the set of Riemannian metrics for which <span>(f_g)</span> is a Morse function is open and dense in the set of all Riemannian metrics.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927661","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
Normalized Ground States for a Fractional Choquard System in $$mathbb {R}$$ $$mathbb {R}$$ 中分数 Choquard 系统的归一化基态
The Journal of Geometric Analysis Pub Date : 2024-05-09 DOI: 10.1007/s12220-024-01629-2
Wenjing Chen, Zexi Wang
{"title":"Normalized Ground States for a Fractional Choquard System in $$mathbb {R}$$","authors":"Wenjing Chen, Zexi Wang","doi":"10.1007/s12220-024-01629-2","DOIUrl":"https://doi.org/10.1007/s12220-024-01629-2","url":null,"abstract":"<p>In this paper, we study the following fractional Choquard system </p><span>$$begin{aligned} begin{aligned} left{ begin{array}{ll} (-Delta )^{1/2}u=lambda _1 u+(I_mu *F(u,v))F_u (u,v), quad text{ in } mathbb {R}, (-Delta )^{1/2}v=lambda _2 v+(I_mu *F(u,v)) F_v(u,v), quad text{ in } mathbb {R}, displaystyle int _{mathbb {R}}|u|^2textrm{d}x=a^2,quad displaystyle int _{mathbb {R}}|v|^2textrm{d}x=b^2,quad u,vin H^{1/2}(mathbb {R}), end{array} right. end{aligned} end{aligned}$$</span><p>where <span>((-Delta )^{1/2})</span> denotes the 1/2-Laplacian operator, <span>(a,b&gt;0)</span> are prescribed, <span>(lambda _1,lambda _2in mathbb {R})</span>, <span>(I_mu (x)=frac{{1}}{{|x|^mu }})</span> with <span>(mu in (0,1))</span>, <span>(F_u,F_v)</span> are partial derivatives of <i>F</i> and <span>(F_u,F_v)</span> have exponential critical growth in <span>(mathbb {R})</span>. By using a minimax principle and analyzing the monotonicity of the ground state energy with respect to the prescribed masses, we obtain at least one normalized ground state solution for the above system.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942133","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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