Monotonicity of first eigenvalues along the Yamabe flow

Liangdi Zhang

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 2, page 387-401
  • ISSN: 0011-4642

Abstract

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We construct some nondecreasing quantities associated to the first eigenvalue of - Δ φ + c R ( c ≥ 1 2 ( n - 2 ) / ( n - 1 ) ) along the Yamabe flow, where Δ φ is the Witten-Laplacian operator with a C 2 function φ . We also prove a monotonic result on the first eigenvalue of - Δ φ + 1 4 ( n / ( n - 1 ) ) R along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of - Δ φ + c R a with a ∈ ( 0 , 1 ) along the Yamabe flow.

How to cite

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Zhang, Liangdi. "Monotonicity of first eigenvalues along the Yamabe flow." Czechoslovak Mathematical Journal 71.2 (2021): 387-401. <http://eudml.org/doc/297520>.

@article{Zhang2021,
abstract = {We construct some nondecreasing quantities associated to the first eigenvalue of $-\Delta _\phi +cR$$(c\ge \frac\{1\}\{2\}(n-2)/(n-1))$ along the Yamabe flow, where $\Delta _\phi $ is the Witten-Laplacian operator with a $C^2$ function $\phi $. We also prove a monotonic result on the first eigenvalue of $-\Delta _\phi + \frac\{1\}\{4\} (n/ (n-1))R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $-\Delta _\phi +cR^a$ with $a\in (0,1)$ along the Yamabe flow.},
author = {Zhang, Liangdi},
journal = {Czechoslovak Mathematical Journal},
keywords = {monotonicity; first eigenvalue; Witten-Laplacian operator; Yamabe flow},
language = {eng},
number = {2},
pages = {387-401},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Monotonicity of first eigenvalues along the Yamabe flow},
url = {http://eudml.org/doc/297520},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Zhang, Liangdi
TI - Monotonicity of first eigenvalues along the Yamabe flow
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 2
SP - 387
EP - 401
AB - We construct some nondecreasing quantities associated to the first eigenvalue of $-\Delta _\phi +cR$$(c\ge \frac{1}{2}(n-2)/(n-1))$ along the Yamabe flow, where $\Delta _\phi $ is the Witten-Laplacian operator with a $C^2$ function $\phi $. We also prove a monotonic result on the first eigenvalue of $-\Delta _\phi + \frac{1}{4} (n/ (n-1))R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $-\Delta _\phi +cR^a$ with $a\in (0,1)$ along the Yamabe flow.
LA - eng
KW - monotonicity; first eigenvalue; Witten-Laplacian operator; Yamabe flow
UR - http://eudml.org/doc/297520
ER -

References

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  1. Cao, X., 10.1007/s00208-006-0043-5, Math. Ann. 337 (2007), 435-441. (2007) Zbl1105.53051MR2262792DOI10.1007/s00208-006-0043-5
  2. Cao, X., 10.1090/S0002-9939-08-09533-6, Proc. Am. Math. Soc. 136 (2008), 4075-4078. (2008) Zbl1166.58007MR2425749DOI10.1090/S0002-9939-08-09533-6
  3. Chow, B., Lu, P., Ni, L., 10.1090/gsm/077, Graduate Studies in Mathematics 77. American Mathematical Society, Providence (2006). (2006) Zbl1118.53001MR2274812DOI10.1090/gsm/077
  4. Fang, S., Xu, H., Zhu, P., 10.1007/s11425-014-4943-7, Sci. China, Math. 58 (2015), 1737-1744. (2015) Zbl1327.53084MR3368179DOI10.1007/s11425-014-4943-7
  5. Fang, S., Yang, F., 10.4134/BKMS.b150530, Bull. Korean Math. Soc. 53 (2016), 1113-1122. (2016) Zbl1350.53051MR3534307DOI10.4134/BKMS.b150530
  6. Fang, S., Yang, F., Zhu, P., 10.1017/S0017089516000537, Glasg. Math. J. 59 (2017), 743-751. (2017) Zbl1408.53088MR3682011DOI10.1017/S0017089516000537
  7. Guo, H., Philipowski, R., Thalmaier, A., 10.2140/pjm.2013.264.61, Pac. J. Math. 264 (2013), 61-81. (2013) Zbl1275.53058MR3079761DOI10.2140/pjm.2013.264.61
  8. Ho, P. T., 10.1007/s10455-018-9608-2, Ann. Glob. Anal. Geom. 54 (2018), 449-472. (2018) Zbl1412.53092MR3878837DOI10.1007/s10455-018-9608-2
  9. Kleiner, B., Lott, J., 10.2140/gt.2008.12.2587, Geom. Topol. 12 (2008), 2587-2858. (2008) Zbl1204.53033MR2460872DOI10.2140/gt.2008.12.2587
  10. Perelman, G., The entropy formula for the Ricci flow and its geometric applications, Available at https://arxiv.org/abs/math/0211159 (2002), 39 pages. (2002) Zbl1130.53001
  11. Reed, M., Simon, B., Methods of Modern Mathematical Physics. IV: Analysis of Operators, Academic Press, New York (1978). (1978) Zbl0401.47001MR0493421
  12. Topping, P., 10.1017/CBO9780511721465, London Mathematical Society Lecture Note Series 325. Cambridge University Press, Cambridge (2006). (2006) Zbl1105.58013MR2265040DOI10.1017/CBO9780511721465
  13. Zhao, L., The first eigenvalue of the Laplace operator under Yamabe flow, Math. Appl. 24 (2011), 274-278. (2011) MR2816261
  14. Zhao, L., 10.1007/s00025-012-0242-1, Result. Math. 63 (2013), 937-948. (2013) Zbl1270.53089MR3057347DOI10.1007/s00025-012-0242-1
  15. Zhao, L., 10.1002/mma.2835, Math. Methods Appl. Sci. 37 (2014), 744-751. (2014) Zbl1288.53068MR3180635DOI10.1002/mma.2835

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