메뉴 건너뛰기




Volumn 66, Issue 3, 2013, Pages 307-331

Heat Flow on Alexandrov Spaces

Author keywords

[No Author keywords available]

Indexed keywords


EID: 84867628882     PISSN: 00103640     EISSN: 10970312     Source Type: Journal    
DOI: 10.1002/cpa.21431     Document Type: Article
Times cited : (121)

References (45)
  • 2
    • 84871647891 scopus 로고    scopus 로고
    • Oxford Lecture Series in Mathematics and Its Applications, 25
    • Oxford University Press, Oxford
    • Ambrosio, L.; Tilli, P. Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and Its Applications, 25. Oxford University Press, Oxford, 2004.
    • (2004) Topics on analysis in metric spaces
    • Ambrosio, L.1    Tilli, P.2
  • 3
    • 0002432396 scopus 로고    scopus 로고
    • New trends in stochastic analysis (Charingworth, 1994)
    • World Scientific Publishing, River Edge, N.J.
    • Bakry, D. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. New trends in stochastic analysis (Charingworth, 1994), pp.43-75. World Scientific Publishing, River Edge, N.J., 1997.
    • (1997) On Sobolev and logarithmic Sobolev inequalities for Markov semigroups , pp. 43-75
    • Bakry, D.1
  • 4
    • 84855544115 scopus 로고    scopus 로고
    • Functional inequalities and Hamilton-Jacobi equations in geodesic spaces
    • no.
    • Balogh, Z. M.; Engoulatov, A.; Hunziker, L.; Maasalo, O. E. Functional inequalities and Hamilton-Jacobi equations in geodesic spaces. Potential Anal. 36 (2012), no.2, 317-337.
    • (2012) Potential Anal. , vol.36 , Issue.2 , pp. 317-337
    • Balogh, Z.M.1    Engoulatov, A.2    Hunziker, L.3    Maasalo, O.E.4
  • 5
    • 50349087918 scopus 로고    scopus 로고
    • Existence and uniqueness of optimal maps on Alexandrov spaces
    • no.
    • Bertrand, J. Existence and uniqueness of optimal maps on Alexandrov spaces. Adv. Math. 219 (2008), no.3, 838-851.
    • (2008) Adv. Math. , vol.219 , Issue.3 , pp. 838-851
    • Bertrand, J.1
  • 7
    • 0003365247 scopus 로고    scopus 로고
    • A course in metric geometry
    • 33. American Mathematical Society, Providence, R.I.
    • Burago, D.; Burago, Yu.; Ivanov, S. A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, R.I., 2001.
    • (2001) Graduate Studies in Mathematics
    • Burago, D.1    Burago, Y.2    Ivanov, S.3
  • 9
    • 0033456306 scopus 로고    scopus 로고
    • Differentiability of Lipschitz functions on metric measure spaces
    • no.
    • Cheeger, J. Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9 (1999), no.3, 428-517.
    • (1999) Geom. Funct. Anal , vol.9 , Issue.3 , pp. 428-517
    • Cheeger, J.1
  • 10
    • 74349096979 scopus 로고    scopus 로고
    • Wasserstein space over the Wiener space
    • no.
    • Fang, S.; Shao, J.; Sturm, K.-T. Wasserstein space over the Wiener space. Probab. Theory Related Fields 146 (2010), no.3-4, 535-565.
    • (2010) Probab. Theory Related Fields , vol.146 , Issue.3-4 , pp. 535-565
    • Fang, S.1    Shao, J.2    Sturm, K.-T.3
  • 11
    • 44349084132 scopus 로고    scopus 로고
    • Absolute continuity of Wasserstein geodesics in the Heisenberg group
    • no.
    • Figalli, A.; Juillet, N. Absolute continuity of Wasserstein geodesics in the Heisenberg group. J. Funct. Anal. 255 (2008), no.1, 133-141.
    • (2008) J. Funct. Anal , vol.255 , Issue.1 , pp. 133-141
    • Figalli, A.1    Juillet, N.2
  • 13
    • 77954423534 scopus 로고    scopus 로고
    • On the heat flow on metric measure spaces: existence, uniqueness and stability
    • no.
    • Gigli, N. On the heat flow on metric measure spaces: existence, uniqueness and stability. Calc. Var. Partial Differential Equations 39 (2010), no.1-2, 101-120.
    • (2010) Calc. Var. Partial Differential Equations , vol.39 , Issue.1-2 , pp. 101-120
    • Gigli, N.1
  • 14
    • 84871778243 scopus 로고    scopus 로고
    • First variation formula in Wasserstein spaces over compact Alexandrov spaces
    • to appear. doi: 10.4153/CMB-2011-110-3
    • Gigli, N.; Ohta, S. First variation formula in Wasserstein spaces over compact Alexandrov spaces. Canad. Math. Bull., to appear. doi: 10.4153/CMB-2011-110-3
    • Canad. Math. Bull
    • Gigli, N.1    Ohta, S.2
  • 15
    • 0032343437 scopus 로고    scopus 로고
    • The variational formulation of the Fokker-Planck equation
    • no.
    • Jordan, R.; Kinderlehrer, D.; Otto, F. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998), no.1, 1-17.
    • (1998) SIAM J. Math. Anal. , vol.29 , Issue.1 , pp. 1-17
    • Jordan, R.1    Kinderlehrer, D.2    Otto, F.3
  • 16
    • 84871659552 scopus 로고    scopus 로고
    • Diffusion by optimal transport in Heisenberg groups
    • Preprint. Available at:
    • Juillet, N. Diffusion by optimal transport in Heisenberg groups. Preprint. Available at: http://www-irma.u-strasbg.fr/∼juillet/papers/grad_2011.pdf
    • Juillet, N.1
  • 17
    • 80955125345 scopus 로고    scopus 로고
    • The Littlewood-Paley-Stein inequality for diffusion processes on general metric spaces
    • no.
    • Kawabi, H.; Miyokawa, T. The Littlewood-Paley-Stein inequality for diffusion processes on general metric spaces. J. Math. Sci. Univ. Tokyo 14 (2007), no.1, 1-30.
    • (2007) J. Math. Sci. Univ. Tokyo , vol.14 , Issue.1 , pp. 1-30
    • Kawabi, H.1    Miyokawa, T.2
  • 18
    • 0000055983 scopus 로고
    • Sobolev spaces and harmonic maps for metric space targets
    • no.
    • Korevaar, N. J.; Schoen, R. M. Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1 (1993), no.3-4, 561-659.
    • (1993) Comm. Anal. Geom , vol.1 , Issue.3-4 , pp. 561-659
    • Korevaar, N.J.1    Schoen, R.M.2
  • 19
    • 77949569321 scopus 로고    scopus 로고
    • Duality on gradient estimates and Wasserstein controls
    • no.
    • Kuwada, K. Duality on gradient estimates and Wasserstein controls. J. Funct. Anal. 258 (2010), no.11, 3758-3774.
    • (2010) J. Funct. Anal. , vol.258 , Issue.11 , pp. 3758-3774
    • Kuwada, K.1
  • 20
    • 0035639753 scopus 로고    scopus 로고
    • Sobolev spaces, Laplacian and heat kernel on Alexandrov spaces
    • no.
    • Kuwae, K.; Machigashira, Y.; Shioya, T. Sobolev spaces, Laplacian and heat kernel on Alexandrov spaces. Math. Z. 238 (2001), no.2, 269-316.
    • (2001) Math. Z. , vol.238 , Issue.2 , pp. 269-316
    • Kuwae, K.1    Machigashira, Y.2    Shioya, T.3
  • 21
    • 0035611176 scopus 로고    scopus 로고
    • On generalized measure contraction property and energy functionals over Lipschitz maps
    • no.
    • Kuwae, K.; Shioya, T. On generalized measure contraction property and energy functionals over Lipschitz maps. Potential Anal. 15 (2001), no.1-2, 105-121.
    • (2001) Potential Anal , vol.15 , Issue.1-2 , pp. 105-121
    • Kuwae, K.1    Shioya, T.2
  • 22
    • 0037287543 scopus 로고    scopus 로고
    • Sobolev spaces and Dirichlet spaces over maps between metric spaces
    • Kuwae, K.; Shioya, T. Sobolev spaces and Dirichlet spaces over maps between metric spaces. J. Reine Angew. Math. 555 (2003), 39-75.
    • (2003) J. Reine Angew. Math. , vol.555 , pp. 39-75
    • Kuwae, K.1    Shioya, T.2
  • 23
    • 0000614531 scopus 로고    scopus 로고
    • The geometry of Markov diffusion generators
    • no.
    • Ledoux, M. The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no.2, 305-366.
    • (2000) Ann. Fac. Sci. Toulouse Math. (6) , vol.9 , Issue.2 , pp. 305-366
    • Ledoux, M.1
  • 24
    • 34548457766 scopus 로고    scopus 로고
    • Hamilton-Jacobi semigroup on length spaces and applications
    • no.
    • Lott, J.; Villani, C. Hamilton-Jacobi semigroup on length spaces and applications. J. Math. Pures Appl. (9) 88 (2007), no.3, 219-229.
    • (2007) J. Math. Pures Appl. (9) , vol.88 , Issue.3 , pp. 219-229
    • Lott, J.1    Villani, C.2
  • 25
    • 33847377614 scopus 로고    scopus 로고
    • Weak curvature conditions and functional inequalities
    • no.
    • Lott, J.; Villani, C. Weak curvature conditions and functional inequalities. J. Funct. Anal. 245 (2007), no.1, 311-333.
    • (2007) J. Funct. Anal. , vol.245 , Issue.1 , pp. 311-333
    • Lott, J.1    Villani, C.2
  • 26
    • 62049083002 scopus 로고    scopus 로고
    • Ricci curvature for metric-measure spaces via optimal transport
    • no.
    • Lott, J.; Villani, C. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 (2009), no.3, 903-991.
    • (2009) Ann. of Math. (2) , vol.169 , Issue.3 , pp. 903-991
    • Lott, J.1    Villani, C.2
  • 27
    • 68149156285 scopus 로고    scopus 로고
    • Gradient flows on Wasserstein spaces over compact Alexandrov spaces
    • no.
    • Ohta, S. Gradient flows on Wasserstein spaces over compact Alexandrov spaces. Amer. J. Math 131 (2009), no.2, 475-516.
    • (2009) Amer. J. Math , vol.131 , Issue.2 , pp. 475-516
    • Ohta, S.1
  • 28
    • 68049121962 scopus 로고    scopus 로고
    • Heat flow on Finsler manifolds
    • no.
    • Ohta, S.; Sturm, K.-T. Heat flow on Finsler manifolds. Comm. Pure. Appl. Math. 62 (2009), no.10, 1386-1433.
    • (2009) Comm. Pure. Appl. Math , vol.62 , Issue.10 , pp. 1386-1433
    • Ohta, S.1    Sturm, K.-T.2
  • 29
    • 84861000915 scopus 로고    scopus 로고
    • Non-contraction of heat flow on Minkowski spaces
    • no.
    • Ohta, S.; Sturm, K.-T. Non-contraction of heat flow on Minkowski spaces. Arch. Ration. Mech. Anal. 204 (2012), no.3, 917-944.
    • (2012) Arch. Ration. Mech. Anal. , vol.204 , Issue.3 , pp. 917-944
    • Ohta, S.1    Sturm, K.-T.2
  • 30
    • 84972500357 scopus 로고
    • The Riemannian structure of Alexandrov spaces
    • no.
    • Otsu, Y.; Shioya, T. The Riemannian structure of Alexandrov spaces. J. Differential Geom. 39 (1994), no.3, 629-658.
    • (1994) J. Differential Geom. , vol.39 , Issue.3 , pp. 629-658
    • Otsu, Y.1    Shioya, T.2
  • 31
    • 0001560970 scopus 로고    scopus 로고
    • The geometry of dissipative evolution equations: the porous medium equation
    • no.
    • Otto, F. The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26 (2001), no.1-2, 101-174.
    • (2001) Comm. Partial Differential Equations , vol.26 , Issue.1-2 , pp. 101-174
    • Otto, F.1
  • 32
    • 0013531142 scopus 로고    scopus 로고
    • DC structure on Alexandrov space with curvature bounded below
    • Available at:
    • Perelman, G. DC structure on Alexandrov space with curvature bounded below. Available at: http://www.math.psu.edu/petrunin/papers/alexandrov/Cstructure.pdf
    • Perelman, G.1
  • 33
    • 15944370025 scopus 로고    scopus 로고
    • Harmonic functions on Alexandrov spaces and their applications
    • Petrunin, A. Harmonic functions on Alexandrov spaces and their applications. Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 135-141.
    • (2003) Electron. Res. Announc. Amer. Math. Soc. , vol.9 , pp. 135-141
    • Petrunin, A.1
  • 34
    • 84867013110 scopus 로고    scopus 로고
    • Alexandrov meets Lott-Villani-Sturm
    • Petrunin, A. Alexandrov meets Lott-Villani-Sturm. Münster J. Math. 4 (2011), 53-64.
    • (2011) Münster J. Math. , vol.4 , pp. 53-64
    • Petrunin, A.1
  • 35
    • 84874948533 scopus 로고    scopus 로고
    • Sharp spectral gap and Li-Yau's estimate on Alexandrov spaces
    • to appear. doi: 10.1007/s00209-012-049-1
    • Qian, Z.; Zhang, H.-C.; Zhu, X.-P. Sharp spectral gap and Li-Yau's estimate on Alexandrov spaces. Math. Z., to appear. doi: 10.1007/s00209-012-049-1
    • Math. Z.
    • Qian, Z.1    Zhang, H.-C.2    Zhu, X.-P.3
  • 36
    • 18444391829 scopus 로고    scopus 로고
    • Poincaré inequality for abstract spaces
    • no.
    • Ranjbar-Motlagh, A. Poincaré inequality for abstract spaces. Bull. Austral. Math. Soc. 71 (2005), no.2, 193-204.
    • (2005) Bull. Austral. Math. Soc. , vol.71 , Issue.2 , pp. 193-204
    • Ranjbar-Motlagh, A.1
  • 37
    • 34547422503 scopus 로고    scopus 로고
    • Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds
    • no.
    • Savaré, G. Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds. C. R. Math. Acad. Sci. Paris 345 (2007), no.3, 151-154.
    • (2007) C. R. Math. Acad. Sci. Paris , vol.345 , Issue.3 , pp. 151-154
    • Savaré, G.1
  • 38
    • 0001581975 scopus 로고    scopus 로고
    • Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality
    • no.
    • Sturm, K.-T. Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75 (1996), no.3, 273-297.
    • (1996) J. Math. Pures Appl. (9) , vol.75 , Issue.3 , pp. 273-297
    • Sturm, K.-T.1
  • 39
    • 33750193237 scopus 로고    scopus 로고
    • On the geometry of metric measure spaces
    • Sturm, K.-T. On the geometry of metric measure spaces. I. Acta Math. 196 (2006), 65-131.
    • (2006) I. Acta Math. , vol.196 , pp. 65-131
    • Sturm, K.-T.1
  • 40
    • 33750180350 scopus 로고    scopus 로고
    • On the geometry of metric measure spaces. II
    • Sturm, K.-T. On the geometry of metric measure spaces. II. Acta Math. 196 (2006), 133-177.
    • (2006) Acta Math. , vol.196 , pp. 133-177
    • Sturm, K.-T.1
  • 41
    • 37349070894 scopus 로고    scopus 로고
    • Grundlehren der mathematischen Wissenschaften, 338
    • Springer, Berlin
    • Villani, C. Optimal transport. Grundlehren der mathematischen Wissenschaften, 338. Springer, Berlin, 2009.
    • (2009) Optimal transport
    • Villani, C.1
  • 42
    • 20744448674 scopus 로고    scopus 로고
    • Transport inequalities, gradient estimates, entropy and Ricci curvature
    • no.
    • von Renesse, M.-K.; Sturm, K.-T. Transport inequalities, gradient estimates, entropy and Ricci curvature. Comm. Pure Appl. Math. 58 (2005), no.7, 923-940.
    • (2005) Comm. Pure Appl. Math. , vol.58 , Issue.7 , pp. 923-940
    • von Renesse, M.-K.1    Sturm, K.-T.2
  • 43
    • 78649372057 scopus 로고    scopus 로고
    • On a new definition of Ricci curvature on Alexandrov spaces
    • no.
    • Zhang, H.-C.; Zhu, X.-P. On a new definition of Ricci curvature on Alexandrov spaces. Acta. Math. Sci. Ser. B Engl. Ed. 30 (2010), no.6, 1949-1974.
    • (2010) Acta. Math. Sci. Ser. B Engl. Ed. , vol.30 , Issue.6 , pp. 1949-1974
    • Zhang, H.-C.1    Zhu, X.-P.2
  • 44
    • 79251520730 scopus 로고    scopus 로고
    • Ricci curvature on Alexandrov spaces and rigidity theorems
    • no.
    • Zhang, H.-C.; Zhu, X.-P. Ricci curvature on Alexandrov spaces and rigidity theorems. Comm. Anal. Geom. 18 (2010), no.3, 503-553.
    • (2010) Comm. Anal. Geom. , vol.18 , Issue.3 , pp. 503-553
    • Zhang, H.-C.1    Zhu, X.-P.2
  • 45
    • 84867000915 scopus 로고    scopus 로고
    • Yau's gradient estimate on Alexandrov spaces
    • no.
    • Zhang, H.-C.; Zhu, X.-P. Yau's gradient estimate on Alexandrov spaces. J. Differential Geom. 91 (2012), no.3, 445-522.
    • (2012) J. Differential Geom. , vol.91 , Issue.3 , pp. 445-522
    • Zhang, H.-C.1    Zhu, X.-P.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.