Lectures on Stability and Constant Scalar Curvature

By Duong H. Phong, Jacob Sturm. 2008

145 citations recorded by the source corpus.

An introduction is provided to some current research trends in stability in geometric invariant theory and the problem of Kaehler metrics of constant scalar curvature. Besides classical notions such as Chow-Mumford stability, the emphasis is on several new stability conditions, such as K-stability, Donaldson's infinite-dimensional GIT, and conditions on the closure of orbits of almost-complex structures under the diffeomorphism group. Related analytic methods are also discussed, including estimates for energy functionals, Tian-Yau-Zelditch approximations, estimates for moment maps, complex Monge-Ampere equations and pluripotential theory, and the Kaehler-Ricci flow

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/71b197e78427779ed987aab932271873d7bf8cf8

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On blowing up extremal Kähler manifolds (2010, background): “9] extended the question to the case of constant scalar curvature metrics and proved the conjecture in the case of toric surfaces [11]. Two recent surveys on this topic are Thomas [32] and PhongSturm [26]. Note that it is essentially known that if an extremal metric exists then the manifold is relat”
Relative K-stability of extremal metrics (2009, background): “We refer to Phong–Sturm [14] for a recent survey and an extensive bibliography.”
Smooth and singular Kahler-Einstein metrics (2014, background): “The interested reader is referred to the numerous existing surveys on related topics [10, 28, 33, 34, 40, 41, 62, 99, 115, 116, 117, 118, 119, 120, 125, 200, 226, 199, 234, 236, 245, 246, 247].”
The Dirichlet problem for degenerate complex Monge-Ampere equations (2009, unknown): “[23, 27] for a survey.”
K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics (2012, methodology): “We will also have use for the following lemma which is an abstract version of the Kempf-Ness approach to the Hilbert-Mumford criterion, used to test stability in Geometric Invariant Theory (compare [46]).”
Geometric pluripotential theory on Kähler manifolds (2019, methodology): “For results about the quantization of the geometry of the space of Kähler metrics, we refer to the original papers [18, 42, 66, 89, 90, 54], as well as the survey [91] along with references therein.”
Existence of approximate Hermitian-Einstein structures on semi-stable bundles (2010, methodology): “The analogue of the Donaldson functional is in this case the Mabuchi K-energy [19], and several analogues of Mumford-Takemoto stability have also been introduced, including Chow-Mumford stability, K-stability (Tian [28], Donaldson [11]), uniform K-stability (Szekelyhidi [27]), slope-stability (Ross-”
Greatest lower bounds on the Ricci curvature of Fano manifolds (2009, methodology): “lity to the existence of constant scalar curvature Kahler metrics, now called the YauTian-Donaldson conjecture, is currently a very active ﬁeld of research. For a survey and many more references, see [19]. In this paper we study Aubin’s [2] continuity method for ﬁnding K¨ahler-Einstein metrics. Give”
K-stability of constant scalar curvature polarization (2008, background): “s also on P∗(W m). We now consider the function (4.7) fm(s) := logkψˆm(exp(s))·Mˆmk CH(H0), s∈ R. Put f˙ m(s) := (dfm/ds)(s). The purpose of this section is to show the following (see Phong and Sturm [20], eqn 7.29, for the leading term; see also [5], p.464–467): 9 Lemma 4.8. Let an be as in (2.3). ”
Fully non-linear parabolic equations on compact Hermitian manifolds (2017, background): “ similar interpretations in terms of stability of the conditions formulated in the previous section. A discussion of stability conditions for constant scalar curvature Ka¨hler metrics can be found in [40]. It would also be very helpful to have a suitable geometric interpretation of conditions such a”