This is my math scrapbook. It includes computations, notes, and worked examples.
Warning: there may be mistakes in these documents; please let me know if you find any!
| Some useful identities, quantities, and inequalities: | ||||
Bochner formula for \( f:M\rightarrow \mathbb{R}\) |
$$ \Delta \frac{1}{2} \langle \nabla f , \nabla f \rangle = \| \nabla ^2 f \| ^2 + Ric(\nabla f , \nabla f) $$ | |||
Hessian of \( \phi : M \rightarrow N \) |
$$ \left( \nabla \left( d \phi\right) \right) _{ij} ^\gamma= \frac{\partial ^2 \phi ^\gamma}{\partial x^i \partial x^j}- ^M \Gamma _{ij} ^k \frac{\partial \phi ^\gamma}{\partial x^k} + ^N \Gamma _{\alpha \beta} ^{\gamma} \frac{\partial \phi ^\alpha}{\partial x^i}\frac{\partial \phi ^\beta}{\partial x^j}$$ | (derivation|tex) |
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Hessian of \( \psi \circ \phi : M\rightarrow N \rightarrow P \) |
$$\nabla d (\psi \circ \phi) = \left( \nabla d \psi \right) (d\phi , d\phi) + d\psi \circ \left( \nabla d \phi \right)$$ | (derivation|tex) |
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Gauss curvature of a surface with metric \(\rho^2 dz \odot d\overline{z}\) |
$$K_\rho (z) = - \frac{\Delta log(\rho(z))}{\rho^2 (z)}$$ | |||
Tension field components of \( \phi:(M,\gamma)\rightarrow (N,g)\) |
$$ \tau (\phi) ^k = \frac{1}{\sqrt{|\gamma_{\alpha \beta} |} } \frac{\partial}{\partial x^\alpha} \left( \gamma ^{\alpha \beta} \sqrt{|\gamma _{\alpha \beta}|} \frac{\partial \phi ^k }{\partial x ^\beta} \right) + \gamma^{\alpha \beta} (\Gamma _{ij} ^{k} \circ \phi) \frac{\partial \phi ^i}{\partial x^\alpha} \frac{\partial \phi ^j}{\partial x^\beta}$$ | (derivation|tex) |
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Euler-Lagrange equations for energy of \( \phi:M\rightarrow N\) |
$$ \tau(\phi) = 0$$ | |||
Harmonicity in conformal coordinates on a surface |
$$ \tau (u) = u_{z \overline{z}} + 2\frac{\rho _u}{\rho} u_z u_\overline{z} =0$$ | (proof|tex) |
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Harmonic maps from surfaces have holomorphic Hopf differential |
$$\tau (u)=0 \Rightarrow \left[\rho ^2 u_z \overline{u_{\overline{z}}}\right]_\overline{z}= 0$$ | (proof|tex) |
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Conformal maps from surfaces \( \equiv \) vanishing Hopf differential |
$$ \Leftrightarrow \rho ^2 u_z \overline{u_{\overline{z}}}\equiv 0$$ | (proof|tex) |
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Monotonicity formula for harmonic maps |
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| Notes compiled from classes/lectures/etc on: | |
| Harmonic maps (from Min-Chun Hong @ Winter School on Geometric PDE) | |
| Higgs bundles (from Steve Bradlow @ Junior GEAR Retreat 2012) | |
| Examples, worked problems, computations: | |
| First Chern class of \( \mathbb{C}P^1 \) | |