Gauge theory and calibrated geometry, I

By G. Tian. 2000

304 citations recorded by the source corpus.

The geometry of submanifolds is intimately related to the theory of functions and vector bundles. It has been of fundamental importance to find out how those two objects interact in many geometric and physical problems. A typical example of this relation is that the Picard group of line bundles on an algebraic manifold is isomorphic to the group of divisors, which is generated by holomorphic hypersurfaces modulo linear equivalence. A similar correspondence can be made between the K-group of sheaves and the Chow ring of holomorphic cycles. There are two more very recent examples of such a relation. The mirror symmetry in string theory has revealed a deeper phenomenon involving special Lagrangian cycles (cf. [SYZ]). On the other hand, C. Taubes has shown that the Seiberg-Witten invariant coincides with the Gromov-Witten invariant on any symplectic 4-manifolds. In this paper, we will show another natural interaction between Yang-Mills connections, which are critical points of a Yang-Mills action associated to a vector bundle, and minimal submanifolds, which have been studied extensively for years in classical differential geometry and the calculus of variations.

View Paper: https://www.semanticscholar.org/paper/2883f5b3d38d578c26ac0a60e721c3ab5690f367

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Citations in the corpus

G2-manifolds and associative submanifolds via semi-Fano 3-folds (2012, background): “[69] G. Tian, Gauge theory and calibrated geometry.”
A partial regularity result for a class of Stationary Yang-Mills Fields in high dimension (2003, background): “2 is a consequence of the epsilon regularity for smooth Yang-Mills Fields established in [10] and [13] once we are able to prove that under the smallness assumption of the curvature density, gauge invariant quantities are smooth.”
Remarks on the geometry of almost complex 6-manifolds (2005, background): “It seems very likely, though I have not tried to verify this, that there would be analogs of the Uhlenbeck removable singularities theorem generalizing to the quasiintegrable case the ones known in the integrable case [7, 21, 22].”
Notes on the octonions (2010, background): “The analytical difficulties one encounters when making this precise are formidable, including non-compactness phenomena in codimension four [32] and two in the gauge theory and submanifold theory respectively.”
Gauge Theory in higher dimensions, II (2009, background): “While considerable progress has been made in this direction by Tian [ 38 ],[32], a detailed theory—in either the gauge theory or submanifold setting—seems still to be fairly distant.”