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Seminar on Geometric Complex Analysis
Seminar information archive ～11/01｜Next seminar｜Future seminars 11/02～
| Date, time & place | Monday 10:30 - 12:00 126Room #126 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Kengo Hirachi, Shigeharu Takayama |
2026/11/30
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Natsuo Miyatake (Tohoku Univ.)
Equilibrium Metrics and Complete Solutions (Japanese)
https://forms.gle/8ERsVDLuKHwbVzm57
Natsuo Miyatake (Tohoku Univ.)
Equilibrium Metrics and Complete Solutions (Japanese)
[ Abstract ]
Let $X$ be a compact Riemann surface. For a positive line bundle $L\to X$, a continuous Hermitian metric $e^{-\phi}h_\ast$ on $L$, and a nonpolar compact subset $K\subseteq X$, one can construct an equilibrium metric from the envelope of subharmonic weight functions bounded above by $\phi$ on $K$. The approximation of equilibrium metrics by sequences of semipositive singular Hermitian metrics has been studied in a variety of contexts, including complex geometry, potential theory, dynamical systems, and probability theory.
In this talk, I will introduce two functions, entropy and free energy, associated with a semipositive singular Hermitian metric on the canonical bundle, an integer $r\geq 2$, and a nonzero real number $\beta$. These quantities are defined by extending the construction, due to Li--Mochizuki, of complete solutions to the Hitchin equation for cyclic Higgs bundles associated with a holomorphic $r$-differential to the setting of complete solutions to a variant of the Hitchin equation associated with semipositive singular Hermitian metrics on the canonical bundle that are not necessarily induced by holomorphic $r$-differentials. Given an equilibrium metric and a sequence of semipositive singular Hermitian metrics approximating it and satisfying a natural normalization condition, I will introduce an appropriate notion of pullback of them to the canonical bundle of a Riemann surface. Within this framework, I will formulate the problem of determining whether the associated entropy increases and the associated free energy decreases. I will initiate the study of this problem and establish several quantitative results. I will also present several examples of approximations of equilibrium metrics to which my results apply, particularly those constructed from Bergman kernels. Furthermore, I will discuss the asymptotic expansion of a lower bound for the entropy.
[ Reference URL ]Let $X$ be a compact Riemann surface. For a positive line bundle $L\to X$, a continuous Hermitian metric $e^{-\phi}h_\ast$ on $L$, and a nonpolar compact subset $K\subseteq X$, one can construct an equilibrium metric from the envelope of subharmonic weight functions bounded above by $\phi$ on $K$. The approximation of equilibrium metrics by sequences of semipositive singular Hermitian metrics has been studied in a variety of contexts, including complex geometry, potential theory, dynamical systems, and probability theory.
In this talk, I will introduce two functions, entropy and free energy, associated with a semipositive singular Hermitian metric on the canonical bundle, an integer $r\geq 2$, and a nonzero real number $\beta$. These quantities are defined by extending the construction, due to Li--Mochizuki, of complete solutions to the Hitchin equation for cyclic Higgs bundles associated with a holomorphic $r$-differential to the setting of complete solutions to a variant of the Hitchin equation associated with semipositive singular Hermitian metrics on the canonical bundle that are not necessarily induced by holomorphic $r$-differentials. Given an equilibrium metric and a sequence of semipositive singular Hermitian metrics approximating it and satisfying a natural normalization condition, I will introduce an appropriate notion of pullback of them to the canonical bundle of a Riemann surface. Within this framework, I will formulate the problem of determining whether the associated entropy increases and the associated free energy decreases. I will initiate the study of this problem and establish several quantitative results. I will also present several examples of approximations of equilibrium metrics to which my results apply, particularly those constructed from Bergman kernels. Furthermore, I will discuss the asymptotic expansion of a lower bound for the entropy.
https://forms.gle/8ERsVDLuKHwbVzm57
