• [1] Thierry Aubin, Meilleures constantes dans le théorème d’inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Functional Analysis 32 (1979), no. 2, 148–174 (French, with English summary). MR 534672, https://doi.org/10.1016/0022-1236(79)90052-1
• [2] D. Bakry, D. Concordet, and M. Ledoux, Optimal heat kernel bounds under logarithmic Sobolev inequalities, ESAIM Probab. Statist. 1 (1995/97), 391–407. MR 1486642, https://doi.org/10.1051/ps:1997115
• [3] Ernst Breitenberger, Uncertainty measures and uncertainty relations for angle observables, Found. Phys. 15 (1985), no. 3, 353–364. MR 811750, https://doi.org/10.1007/BF00737323
• [4] Huai-Dong Cao, Recent progress on Ricci solitons, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 1–38. MR 2648937
• [5] Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175–185. MR 2732975
• [6] L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259–275. MR 768824
• [7] Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207
• [8] Manfredo Perdigão do Carmo and Changyu Xia, Complete manifolds with non-negative Ricci curvature and the Caffarelli-Kohn-Nirenberg inequalities, Compos. Math. 140 (2004), no. 3, 818–826. MR 2041783, https://doi.org/10.1112/S0010437X03000745
• [9] Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406–480. MR 1484888
• [10] Bing-Long Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363–382. MR 2520796
• [11] Bennett Chow, Peng Lu, and Bo Yang, Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons, C. R. Math. Acad. Sci. Paris 349 (2011), no. 23-24, 1265–1267 (English, with English and French summaries). MR 2861997, https://doi.org/10.1016/j.crma.2011.11.004
• [12] L. Cohen, Time-Frequency analysis: theory and applications, Prentice Hall, 1995.
• [13] Pei Dang, Guan-Tie Deng, and Tao Qian, A sharper uncertainty principle, J. Funct. Anal. 265 (2013), no. 10, 2239–2266. MR 3091814, https://doi.org/10.1016/j.jfa.2013.07.023
• [14] Olivier Druet, Emmanuel Hebey, and Michel Vaugon, Optimal Nash’s inequalities on Riemannian manifolds: the influence of geometry, Internat. Math. Res. Notices 14 (1999), 735–779. MR 1704184, https://doi.org/10.1155/S1073792899000380
• [15] W. Erb, Uncertainty principles on Riemannian manifolds, PhD Thesis, 2010.
• [16] Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, https://doi.org/10.1090/S0273-0979-1983-15154-6
• [17] Manuel Fernández-López and Eduardo García-Río, On gradient Ricci solitons with constant scalar curvature, Proc. Amer. Math. Soc. 144 (2016), no. 1, 369–378. MR 3415603, https://doi.org/10.1090/S0002-9939-2015-12693-7
• [18] Gerald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. MR 1448337, https://doi.org/10.1007/BF02649110
• [19] Mikhail Feldman, Tom Ilmanen, and Dan Knopf, Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons, J. Differential Geom. 65 (2003), no. 2, 169–209. MR 2058261
• [20] Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
• [21] Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1688256
• [22] W. Heisenberg, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Physik 43 (1927), 172-198.
• [23] L. Huang, A. Kristály and W. Zhao Sharp uncertainty principles on general Finsler manifolds, accepted, Trans. Amer. Math. Soc., DOI: https://doi.org/10.1090/tran/8178.
• [24] Ismail Kombe and Murad Özaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6191–6203. MR 2538592, https://doi.org/10.1090/S0002-9947-09-04642-X
• [25] Ismail Kombe and Murad Özaydin, Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5035–5050. MR 3074365, https://doi.org/10.1090/S0002-9947-2013-05763-7
• [26] Alexandru Kristály, Metric measure spaces supporting Gagliardo-Nirenberg inequalities: volume non-collapsing and rigidities, Calc. Var. Partial Differential Equations 55 (2016), no. 5, Art. 112, 27. MR 3549918, https://doi.org/10.1007/s00526-016-1065-9
• [27] Alexandru Kristály, Sharp uncertainty principles on Riemannian manifolds: the influence of curvature, J. Math. Pures Appl. (9) 119 (2018), 326–346 (English, with English and French summaries). MR 3862150, https://doi.org/10.1016/j.matpur.2017.09.002
• [28] Alexandru Kristály and Shin-ichi Ohta, Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications, Math. Ann. 357 (2013), no. 2, 711–726. MR 3096522, https://doi.org/10.1007/s00208-013-0918-1
• [29] M. Ledoux, On manifolds with non-negative Ricci curvature and Sobolev inequalities, Comm. Anal. Geom. 7 (1999), no. 2, 347–353. MR 1685586, https://doi.org/10.4310/CAG.1999.v7.n2.a7
• [30] Peter Li and Jiaping Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 6, 921–982 (English, with English and French summaries). MR 2316978, https://doi.org/10.1016/j.ansens.2006.11.001
• [31] Vincent Minerbe, Weighted Sobolev inequalities and Ricci flat manifolds, Geom. Funct. Anal. 18 (2009), no. 5, 1696–1749. MR 2481740, https://doi.org/10.1007/s00039-009-0701-3
• [32] S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401–404. MR 4518
• [33] V. H. Nguyen, Sharp Caffarelli-Kohn-Nirenberg inequalities on Riemannian manifolds: the influence of curvature, (2017), arXiv:1709.06120 [math.FA].
• [34] Stefano Pigola, Michele Rimoldi, and Alberto G. Setti, Remarks on non-compact gradient Ricci solitons, Math. Z. 268 (2011), no. 3-4, 777–790. MR 2818729, https://doi.org/10.1007/s00209-010-0695-4
• [35] Hermann Weyl, The theory of groups and quantum mechanics, Dover Publications, Inc., New York, 1950. Translated from the second (revised) German edition by H. P. Robertson; Reprint of the 1931 English translation. MR 3363447
• [36] Changyu Xia, Complete manifolds with nonnegative Ricci curvature and almost best Sobolev constant, Illinois J. Math. 45 (2001), no. 4, 1253–1259. MR 1894894
• [37] Changyu Xia, The Gagliardo-Nirenberg inequalities and manifolds of non-negative Ricci curvature, J. Funct. Anal. 224 (2005), no. 1, 230–241. MR 2139111, https://doi.org/10.1016/j.jfa.2004.11.009
• [38] Changyu Xia, The Caffarelli-Kohn-Nirenberg inequalities on complete manifolds, Math. Res. Lett. 14 (2007), no. 5, 875–885. MR 2350131, https://doi.org/10.4310/MRL.2007.v14.n5.a14

References

[1]

Thierry Aubin, Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Functional Analysis 32 (1979), no. 2, 148-174 (French, with English summary). MR 534672, https://doi.org/10.1016/0022-1236(79)90052-1

[2]

D. Bakry, D. Concordet, and M. Ledoux, Optimal heat kernel bounds under logarithmic Sobolev inequalities, ESAIM Probab. Statist. 1 (1995/97), 391-407. MR 1486642, https://doi.org/10.1051/ps:1997115

[3]

Ernst Breitenberger, Uncertainty measures and uncertainty relations for angle observables, Found. Phys. 15 (1985), no. 3, 353-364. MR 811750, https://doi.org/10.1007/BF00737323

[4]

Huai-Dong Cao, Recent progress on Ricci solitons, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 1-38. MR 2648937

[5]

Huai-Dong Cao and Detang Zhou, On complete gradient shrinking Ricci solitons, J. Differential Geom. 85 (2010), no. 2, 175-185. MR 2732975

[6]

L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259-275. MR 768824

[7]

Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207

[8]

Manfredo Perdigão do Carmo and Changyu Xia, Complete manifolds with non-negative Ricci curvature and the Caffarelli-Kohn-Nirenberg inequalities, Compos. Math. 140 (2004), no. 3, 818-826. MR 2041783, https://doi.org/10.1112/S0010437X03000745

[9]

Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406-480. MR 1484888

[10]

Bing-Long Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363-382. MR 2520796

[11]

Bennett Chow, Peng Lu, and Bo Yang, Lower bounds for the scalar curvatures of noncompact gradient Ricci solitons, C. R. Math. Acad. Sci. Paris 349 (2011), no. 23-24, 1265-1267 (English, with English and French summaries). MR 2861997, https://doi.org/10.1016/j.crma.2011.11.004

[12]

L. Cohen, Time-Frequency analysis: theory and applications, Prentice Hall, 1995.

[13]

Pei Dang, Guan-Tie Deng, and Tao Qian, A sharper uncertainty principle, J. Funct. Anal. 265 (2013), no. 10, 2239-2266. MR 3091814, https://doi.org/10.1016/j.jfa.2013.07.023

[14]

Olivier Druet, Emmanuel Hebey, and Michel Vaugon, Optimal Nash's inequalities on Riemannian manifolds: the influence of geometry, Internat. Math. Res. Notices 14 (1999), 735-779. MR 1704184, https://doi.org/10.1155/S1073792899000380

[15]

W. Erb, Uncertainty principles on Riemannian manifolds, PhD Thesis, 2010.

[16]

Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129-206. MR 707957, https://doi.org/10.1090/S0273-0979-1983-15154-6

[17]

Manuel Fernández-López and Eduardo García-Río, On gradient Ricci solitons with constant scalar curvature, Proc. Amer. Math. Soc. 144 (2016), no. 1, 369-378. MR 3415603, https://doi.org/10.1090/proc/12693

[18]

Gerald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207-238. MR 1448337, https://doi.org/10.1007/BF02649110

[19]

Mikhail Feldman, Tom Ilmanen, and Dan Knopf, Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons, J. Differential Geom. 65 (2003), no. 2, 169-209. MR 2058261

[20]

Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7-136. MR 1375255

[21]

Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. MR 1688256

[22]

W. Heisenberg, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Physik 43 (1927), 172-198.

[23]

L. Huang, A. Kristály and W. Zhao Sharp uncertainty principles on general Finsler manifolds, accepted, Trans. Amer. Math. Soc., DOI: https://doi.org/10.1090/tran/8178.

[24]

Ismail Kombe and Murad Özaydin, Improved Hardy and Rellich inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6191-6203. MR 2538592, https://doi.org/10.1090/S0002-9947-09-04642-X

[25]

Ismail Kombe and Murad Özaydin, Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5035-5050. MR 3074365, https://doi.org/10.1090/S0002-9947-2013-05763-7

[26]

Alexandru Kristály, Metric measure spaces supporting Gagliardo-Nirenberg inequalities: volume non-collapsing and rigidities, Calc. Var. Partial Differential Equations 55 (2016), no. 5, Art. 112, 27. MR 3549918, https://doi.org/10.1007/s00526-016-1065-9

[27]

Alexandru Kristály, Sharp uncertainty principles on Riemannian manifolds: the influence of curvature, J. Math. Pures Appl. (9) 119 (2018), 326-346 (English, with English and French summaries). MR 3862150, https://doi.org/10.1016/j.matpur.2017.09.002

[28]

Alexandru Kristály and Shin-ichi Ohta, Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications, Math. Ann. 357 (2013), no. 2, 711-726. MR 3096522, https://doi.org/10.1007/s00208-013-0918-1

[29]

M. Ledoux, On manifolds with non-negative Ricci curvature and Sobolev inequalities, Comm. Anal. Geom. 7 (1999), no. 2, 347-353. MR 1685586, https://doi.org/10.4310/CAG.1999.v7.n2.a7

[30]

Peter Li and Jiaping Wang, Weighted Poincaré inequality and rigidity of complete manifolds, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 6, 921-982 (English, with English and French summaries). MR 2316978, https://doi.org/10.1016/j.ansens.2006.11.001

[31]

Vincent Minerbe, Weighted Sobolev inequalities and Ricci flat manifolds, Geom. Funct. Anal. 18 (2009), no. 5, 1696-1749. MR 2481740, https://doi.org/10.1007/s00039-009-0701-3

[32]

S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401-404. MR 4518

[33]

V. H. Nguyen, Sharp Caffarelli-Kohn-Nirenberg inequalities on Riemannian manifolds: the influence of curvature, (2017), arXiv:1709.06120 [math.FA].

[34]

Stefano Pigola, Michele Rimoldi, and Alberto G. Setti, Remarks on non-compact gradient Ricci solitons, Math. Z. 268 (2011), no. 3-4, 777-790. MR 2818729, https://doi.org/10.1007/s00209-010-0695-4

[35]

Hermann Weyl, The theory of groups and quantum mechanics, Dover Publications, Inc., New York, 1950. Translated from the second (revised) German edition by H. P. Robertson; Reprint of the 1931 English translation. MR 3363447

[36]

Changyu Xia, Complete manifolds with nonnegative Ricci curvature and almost best Sobolev constant, Illinois J. Math. 45 (2001), no. 4, 1253-1259. MR 1894894

[37]

Changyu Xia, The Gagliardo-Nirenberg inequalities and manifolds of non-negative Ricci curvature, J. Funct. Anal. 224 (2005), no. 1, 230-241. MR 2139111, https://doi.org/10.1016/j.jfa.2004.11.009

[38]

Changyu Xia, The Caffarelli-Kohn-Nirenberg inequalities on complete manifolds, Math. Res. Lett. 14 (2007), no. 5, 875-885. MR 2350131, https://doi.org/10.4310/MRL.2007.v14.n5.a14