In this paper, we introduce bi-slant Riemannian maps from Riemannian manifolds to Kenmotsu manifolds, which are the natural generalizations of invariant, anti-invariant, semi-invariant, slant, semi-slant and hemi-slant Riemannian maps, with nontrivial examples. We study these maps and give some curvature relations for $(rangeF_*)^perp$. We construct Chen-Ricci inequalities, DDVV inequalities, and further some optimal inequalities involving Casorati curvatures from bi-slant Riemannian manifolds to Kenmotsu space forms.
{"title":"Bi-slant Riemannian maps to Kenmotsu manifolds and some optimal inequalities","authors":"Adeeba Zaidi, Gauree Shanker","doi":"arxiv-2409.01636","DOIUrl":"https://doi.org/arxiv-2409.01636","url":null,"abstract":"In this paper, we introduce bi-slant Riemannian maps from Riemannian\u0000manifolds to Kenmotsu manifolds, which are the natural generalizations of\u0000invariant, anti-invariant, semi-invariant, slant, semi-slant and hemi-slant\u0000Riemannian maps, with nontrivial examples. We study these maps and give some\u0000curvature relations for $(rangeF_*)^perp$. We construct Chen-Ricci\u0000inequalities, DDVV inequalities, and further some optimal inequalities\u0000involving Casorati curvatures from bi-slant Riemannian manifolds to Kenmotsu\u0000space forms.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199065","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}
Alvarez-Consul--Garcia-Fernandez--Garcia-Prada introduced the K"ahler-Yang-Mills equations. They also introduced the $alpha$-Futaki character, an analog of the Futaki invariant, as an obstruction to the existence of the K"ahler-Yang-Mills equations. The equations depend on a coupling constant $alpha$. Solutions of these equations with coupling constant $alpha>0$ are of utmost importance. In this paper, we provide a formula for the $alpha$-Futaki character on certain ample line bundles over toric manifolds. We then show that there are no solutions with $alpha>0$ on certain ample line bundles over certain toric manifolds and compute the value of $alpha$ if a solution exists. We also relate our result to the existence result of Keller-Friedman in dimension-two.
{"title":"A formula for the α-Futaki character","authors":"Kartick Ghosh","doi":"arxiv-2409.01734","DOIUrl":"https://doi.org/arxiv-2409.01734","url":null,"abstract":"Alvarez-Consul--Garcia-Fernandez--Garcia-Prada introduced the\u0000K\"ahler-Yang-Mills equations. They also introduced the $alpha$-Futaki\u0000character, an analog of the Futaki invariant, as an obstruction to the\u0000existence of the K\"ahler-Yang-Mills equations. The equations depend on a\u0000coupling constant $alpha$. Solutions of these equations with coupling constant\u0000$alpha>0$ are of utmost importance. In this paper, we provide a formula for\u0000the $alpha$-Futaki character on certain ample line bundles over toric\u0000manifolds. We then show that there are no solutions with $alpha>0$ on certain\u0000ample line bundles over certain toric manifolds and compute the value of\u0000$alpha$ if a solution exists. We also relate our result to the existence\u0000result of Keller-Friedman in dimension-two.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199051","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}
It is well-known that every cuspidal edge in the Euclidean space E^3 cannot have a bounded mean curvature function. On the other hand, in the Lorentz-Minkowski space L^3, zero mean curvature surfaces admit cuspidal edges. One natural question is to ask when a cuspidal edge has bounded mean curvature in L^3. We show that such a phenomenon occurs only when the image of the singular set is a light-like curve in L^3. Moreover, we also investigate the behavior of principal curvatures in this case as well as other possible cases. In this paper, almost all calculations are given for generalized cuspidal edges as well as for cuspidal edges. We define the "order" at each generalized cuspidal edge singular point is introduced. As nice classes of zero-mean curvature surfaces in L^3,"maxfaces" and "minfaces" are known, and generalized cuspidal edge singular points on maxfaces and minfaces are of order four. One of the important results is that the generalized cuspidal edges of order four exhibit a quite similar behaviors as those on maxfaces and minfaces.
{"title":"Cuspidal edges and generalized cuspidal edges in the Lorentz-Minkowski 3-space","authors":"T. Fukui, R. Kinoshita, D. Pei, M. Umehara, H. Yu","doi":"arxiv-2409.01603","DOIUrl":"https://doi.org/arxiv-2409.01603","url":null,"abstract":"It is well-known that every cuspidal edge in the Euclidean space E^3 cannot\u0000have a bounded mean curvature function. On the other hand, in the\u0000Lorentz-Minkowski space L^3, zero mean curvature surfaces admit cuspidal edges.\u0000One natural question is to ask when a cuspidal edge has bounded mean curvature\u0000in L^3. We show that such a phenomenon occurs only when the image of the\u0000singular set is a light-like curve in L^3. Moreover, we also investigate the\u0000behavior of principal curvatures in this case as well as other possible cases.\u0000In this paper, almost all calculations are given for generalized cuspidal edges\u0000as well as for cuspidal edges. We define the \"order\" at each generalized\u0000cuspidal edge singular point is introduced. As nice classes of zero-mean\u0000curvature surfaces in L^3,\"maxfaces\" and \"minfaces\" are known, and generalized\u0000cuspidal edge singular points on maxfaces and minfaces are of order four. One\u0000of the important results is that the generalized cuspidal edges of order four\u0000exhibit a quite similar behaviors as those on maxfaces and minfaces.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199052","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}
In this paper we extend Alexandrov's sphere theorems for higher-order mean curvature functions to $ W^{2,n} $-regular hypersurfaces under a general degenerate elliptic condition. The proof is based on the extension of the Montiel-Ros argument to the aforementioned class of hypersurfaces and on the existence of suitable Legendrian cycles over them. Using the latter we can also prove that there are $ n $-dimensional Legendrian cycles with $ 2n $-dimensional support, hence answering a question by Rataj and Zaehle. Finally we provide a very general version of the umbilicality theorem for Sobolev-type hypersurfaces.
{"title":"Alexandrov sphere theorems for $ W^{2,n} $-hypersurfaces","authors":"Mario Santilli, Paolo Valentini","doi":"arxiv-2409.01061","DOIUrl":"https://doi.org/arxiv-2409.01061","url":null,"abstract":"In this paper we extend Alexandrov's sphere theorems for higher-order mean\u0000curvature functions to $ W^{2,n} $-regular hypersurfaces under a general\u0000degenerate elliptic condition. The proof is based on the extension of the\u0000Montiel-Ros argument to the aforementioned class of hypersurfaces and on the\u0000existence of suitable Legendrian cycles over them. Using the latter we can also\u0000prove that there are $ n $-dimensional Legendrian cycles with $ 2n\u0000$-dimensional support, hence answering a question by Rataj and Zaehle. Finally\u0000we provide a very general version of the umbilicality theorem for Sobolev-type\u0000hypersurfaces.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199053","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}
Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, Felix Schulze
Bamler--Kleiner recently proved a multiplicity-one theorem for mean curvature flow in $mathbb{R}^3$ and combined it with the authors' work on generic mean curvature flows to fully resolve Huisken's genericity conjecture. In this paper we show that a short density-drop theorem plus the Bamler--Kleiner multiplicity-one theorem for tangent flows at the first nongeneric singular time suffice to resolve Huisken's conjecture -- without relying on the strict genus drop theorem for one-sided ancient flows previously established by the authors.
{"title":"Revisiting generic mean curvature flow in $mathbb{R}^3$","authors":"Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis, Felix Schulze","doi":"arxiv-2409.01463","DOIUrl":"https://doi.org/arxiv-2409.01463","url":null,"abstract":"Bamler--Kleiner recently proved a multiplicity-one theorem for mean curvature\u0000flow in $mathbb{R}^3$ and combined it with the authors' work on generic mean\u0000curvature flows to fully resolve Huisken's genericity conjecture. In this paper\u0000we show that a short density-drop theorem plus the Bamler--Kleiner\u0000multiplicity-one theorem for tangent flows at the first nongeneric singular\u0000time suffice to resolve Huisken's conjecture -- without relying on the strict\u0000genus drop theorem for one-sided ancient flows previously established by the\u0000authors.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199054","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}
After quick survey of some key results and open questions about the structure of singularities of minimal surfaces, we discuss recent work~cite{Sim23} on singularities of stable minimal hypersurfaces, including some simplifications of the main technical discussion in~cite{Sim23}.
{"title":"Singularities of minimal submanifolds","authors":"Leon Simon","doi":"arxiv-2409.00928","DOIUrl":"https://doi.org/arxiv-2409.00928","url":null,"abstract":"After quick survey of some key results and open questions about the structure\u0000of singularities of minimal surfaces, we discuss recent work~cite{Sim23} on\u0000singularities of stable minimal hypersurfaces, including some simplifications\u0000of the main technical discussion in~cite{Sim23}.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199064","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}
We run the continuity method for Mabuchi's generalization of K"{a}hler-Einstein metrics, assuming the existence of an extremal K"{a}hler metric. It gives an analytic proof (without minimal model program) of the recent existence result obtained by Apostolov, Lahdili and Nitta. Our key observation is the boundedness of the energy functionals along the continuity method. The same argument can be applied to general $g$-solitons and $g$-extremal metrics.
{"title":"Continuity method for the Mabuchi soliton on the extremal Fano manifolds","authors":"Tomoyuki Hisamoto, Satoshi Nakamura","doi":"arxiv-2409.00886","DOIUrl":"https://doi.org/arxiv-2409.00886","url":null,"abstract":"We run the continuity method for Mabuchi's generalization of\u0000K\"{a}hler-Einstein metrics, assuming the existence of an extremal K\"{a}hler\u0000metric. It gives an analytic proof (without minimal model program) of the\u0000recent existence result obtained by Apostolov, Lahdili and Nitta. Our key\u0000observation is the boundedness of the energy functionals along the continuity\u0000method. The same argument can be applied to general $g$-solitons and\u0000$g$-extremal metrics.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199066","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}
We provide new type of decay estimate for scalar curvatures of steady gradient Ricci solitons. We also give certain upper bound for the diameter of a Riemannian manifold whose $infty$-Bakry--Emery Ricci tensor is bounded by some positive constant from below. For the proofs, we use $mu$-bubbles introduced by Gromov.
{"title":"Notes on scalar curvature lower bounds of steady gradient Ricci solitons","authors":"Shota Hamanaka","doi":"arxiv-2409.00583","DOIUrl":"https://doi.org/arxiv-2409.00583","url":null,"abstract":"We provide new type of decay estimate for scalar curvatures of steady\u0000gradient Ricci solitons. We also give certain upper bound for the diameter of a\u0000Riemannian manifold whose $infty$-Bakry--Emery Ricci tensor is bounded by some\u0000positive constant from below. For the proofs, we use $mu$-bubbles introduced\u0000by Gromov.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199067","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}
We consider certain fiber bundles over a paraquaternionic contact manifolds, called twistor and reflector spaces, and show that these carry an intrinsic geometric structure that is always integrable.
{"title":"Twistor and Reflector spaces for paraquaternionic contact manifolds","authors":"Stefan Ivanov, Ivan Minchev, Marina Tchomakova","doi":"arxiv-2409.00539","DOIUrl":"https://doi.org/arxiv-2409.00539","url":null,"abstract":"We consider certain fiber bundles over a paraquaternionic contact manifolds,\u0000called twistor and reflector spaces, and show that these carry an intrinsic\u0000geometric structure that is always integrable.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199068","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}
We consider a general class of elliptic equations on hypercomplex manifolds which includes the quaternionic Monge-Amp`ere equation, the quaternionic Hessian equation and the Monge-Amp`ere equation for quaternionic $(n-1)$-plurisubharmonic functions. We prove that under suitable assumptions the solutions to these equations on hyperk"ahler manifolds satisfy a $C^{2,alpha}$ a priori estimate.
{"title":"Fully non-linear elliptic equations on compact hyperkähler manifolds","authors":"Giovanni Gentili, Luigi Vezzoni","doi":"arxiv-2409.00420","DOIUrl":"https://doi.org/arxiv-2409.00420","url":null,"abstract":"We consider a general class of elliptic equations on hypercomplex manifolds\u0000which includes the quaternionic Monge-Amp`ere equation, the quaternionic\u0000Hessian equation and the Monge-Amp`ere equation for quaternionic\u0000$(n-1)$-plurisubharmonic functions. We prove that under suitable assumptions\u0000the solutions to these equations on hyperk\"ahler manifolds satisfy a\u0000$C^{2,alpha}$ a priori estimate.","PeriodicalId":501113,"journal":{"name":"arXiv - MATH - Differential Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199070","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}
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