We concentrate on using the traceless Ricci tensor and the Bochner curvature tensor to study the rigidity problems for complete Kähler manifolds. We derive some elliptic differential inequalities from Weitzenbock formulas for the traceless Ricci tensor of Kähler manifolds with constant scalar curvature and the Bochner tensor of Kähler-Einstein manifolds respectively. Using elliptic estimates and maximum principle, several Lp and L∞ pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature and complex space forms among Kähler-Einstein manifolds. Our results can be regarded as a complex analogues to the rigidity results for Riemannian manifolds. Moreover, our main results especially establish the rigidity theorems for complete noncompact Kähler manifolds and noncompact Kähler-Einstein manifolds under some pointwise pinching conditions or global integral pinching conditions. To the best of our knowledge, these kinds of results have not been reported.
Tian CHONG
,
Yuxin DONG
,
Hezi LIN
,
Yibin REN
. RIGIDITY THEOREMS OF COMPLETE KÄHLER-EINSTEIN MANIFOLDS AND COMPLEX SPACE FORMS[J]. Acta mathematica scientia, Series B, 2019
, 39(2)
: 339
-356
.
DOI: 10.1007/s10473-019-0201-y
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