One of my major research interests is in the Schwarz lemma, and more generally, the Bochner technique.
The Bochner technique is analytic method for obtaining vanishing and estimation theorems on topological or geometric invariants (e.g., Betti numbers, Hodge numbers, norms of derivatives of holomorphic maps) on a closed (compact without boundary) manifold under curvature assumptions. The Schwarz lemma is the Bochner technique applied to the squared norm of the derivative of a holomorphic map. In the context of harmonic maps, this is called the energy density.
The Schwarz Lemma.
I surveyed many of the recent developments on the Schwarz lemma in my paper:
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The Schwarz Lemma: An Odyssey. The Rocky Mountain Journal of Mathematics, 52, no. 4, pp. 1141--1155, (2022). PDF ᐧ arXiv ᐧ Journal
The above survey leads up to the Hermitian Aubin--Yau inequality that I proved in:
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The Schwarz lemma in Kähler and non-Kähler geometry. Asian Journal of Mathematics, 27, no. 1, pp. 121-134, (2023). PDF ᐧ arXiv ᐧ Journal
The pursuit of a Hermitian Aubin--Yau inequality (and even this terminology) comes from my discussions with Yanir Rubinstein. In particular, Yanir's perspective on the various incarnations of the Schwarz lemma that are seen in his paper Smooth and Singular Kähler--Einstein metrics, was very influentional.
I applied the Schwarz lemma to problems to long-time solutions of the Kähler--Ricci flow in my paper:
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Second-order estimates for collapsed limits of Ricci-flat Kähler metrics. Canadian Mathematics Bulletin, 66, no. 3, pp. 912-926, (2023). PDF ᐧ arXiv ᐧ Journal
With James Stanfield, I extended Royden's Schwarz Lemma (1980) to holomorphic maps into pluriclosed manifolds.
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A general Schwarz lemma for Hermitian manifolds. Joint with James Stanfield. PDF ᐧ arXiv
I spoke about the results of the above paper in the following talks:
- Invariant Metrics in Complex Analysis and a Conjecture of Kobayashi and Lang. The University of Melbourne Topology Seminar, April 29, 2024. Slides
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Invariant Metrics in Complex Analysis and a Conjecture of Kobayashi and Lang. Virtual Seminar on Geometry with Symmetries, March 27, 2024. Slides ᐧ Video
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A General Schwarz Lemma for Hermitian Manifolds with Applications to a Conjecture of Kobayashi and Lang. Generalized Ricci flow seminar, February 20, 2024. Slides ᐧ Video
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Curvature aspects of hyperbolicity in complex geometry. Complex Analysis, Geometry, and Dynamics; Portorož, June 5, 2023. Slides
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Recent developments on the Schwarz lemma. The University of Rome "Tor Vergatta", June 3, 2023. Slides
Locality problems, Einstein metrics, and constant holomorphic sectional curvature.
Artem Pulemotov and I discovered a new application for the Schwarz lemma in our paper:
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Hermitian metrics with vanishing second Chern Ricci curvature. Joint with Artem Pulemotov, Bulletin of the London Mathematical Society. PDF ᐧ arXiv ᐧ Journal
I spoke about the present and future applications of this new technique to locality problems in my talks:
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A Locality Theorem for Einstein metrics on compact complex manifolds. Annual meeting of the Australian Mathematical Society, Topology session, December 8, 2023. Slides
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A Locality Theorem for Einstein metrics on compact complex manifolds. Annual meeting of the Australian Mathematical Society, Geometric Analysis session, December 5, 2023. Slides
My works with Artem Pulemotov and James Stanfield
were applied to the constant holomorphic sectional
curvature problem in my paper with Kai Tang:
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On Hermitian manifolds with vanishing curvature. Joint with Kai Tang, Mathematische Zeitschrift, 309, no. 12, 14 pp., (2024). PDF ᐧ arXiv ᐧ Journal
Vanishing theorems for Hodge numbers.
The Schwarz lemma, which has served as the backbone of my research, is one manifestation of the Bochner technique. Peter Petersen and Matthias Wink (2021) recently introduced a new approach to this technique, resulting in significant improvements on existing vanishing theorems for the Betti numbers of a Riemannian manifold.
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Matthias spoke of these results at the Virtual Seminar on Geometry with Symmetries. Video.
This new approach was subsequently applied to the curvature operator of Kähler manifolds to obtain information on Hodge numbers. They achieved optimal results for certain Hodge numbers, but the results were far from optimal in several cases, revealing a subtle obstruction.
These obstructions have been overcome and will appear in the forthcoming article:
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Vanishing theorems for Hodge numbers and the Calabi curvature operator. Joint with Jan Nienhaus, Peter Petersen, James Stanfield, and Matthias Wink. In preparation.
Peter Petersen has already given a number of talks on the results: