File: Mathlib/Geometry/Manifold/Complex.lean

full_name: Complex.norm_eventually_eq_of_mdifferentiableAt_of_isLocalMax
code: /-- **Maximum modulus principle**: if `f : M → F` is complex differentiable in a neighborhood of `c`
and the norm `‖f z‖` has a local maximum at `c`, then `‖f z‖` is locally constant in a neighborhood
of `c`. This is a manifold version of `Complex.norm_eventually_eq_of_isLocalMax`. -/
theorem Complex.norm_eventually_eq_of_mdifferentiableAt_of_isLocalMax {f : M → F} {c : M}
    (hd : ∀ᶠ z in 𝓝 c, MDifferentiableAt I 𝓘(ℂ, F) f z) (hc : IsLocalMax (norm ∘ f) c) :
    ∀ᶠ y in 𝓝 c, ‖f y‖ = ‖f c‖ 
goal: ∀ᶠ (y : M) in 𝓝 c, ‖f y‖ = ‖f c‖

full_name: MDifferentiableOn.norm_eqOn_of_isPreconnected_of_isMaxOn
code: /-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space. Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `‖f x‖ = ‖f c‖` for all `x ∈ U`. -/
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : M → F} {U : Set M} {c : M}
    (hd : MDifferentiableOn I 𝓘(ℂ, F) f U) (hc : IsPreconnected U) (ho : IsOpen U)
    (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const M ‖f c‖) U 
goal: Set.EqOn (norm ∘ f) (Function.const M ‖f c‖) U

full_name: MDifferentiableOn.eqOn_of_isPreconnected_of_isMaxOn_norm
code: /-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a
complex normed space.  Let `f : E → F` be a function that is complex differentiable on `U`. Suppose
that `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `f x = f c` for all `x ∈ U`.

TODO: change assumption from `IsMaxOn` to `IsLocalMax`. -/
theorem eqOn_of_isPreconnected_of_isMaxOn_norm [StrictConvexSpace ℝ F] {f : M → F} {U : Set M}
    {c : M} (hd : MDifferentiableOn I 𝓘(ℂ, F) f U) (hc : IsPreconnected U) (ho : IsOpen U)
    (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c) : EqOn f (const M (f c)) U 
goal: Set.EqOn f (Function.const M (f c)) U

full_name: MDifferentiableOn.apply_eq_of_isPreconnected_isCompact_isOpen
code: /-- If a function `f : M → F` from a complex manifold to a complex normed space is holomorphic on a
(pre)connected compact open set, then it is a constant on this set. -/
theorem apply_eq_of_isPreconnected_isCompact_isOpen {f : M → F} {U : Set M} {a b : M}
    (hd : MDifferentiableOn I 𝓘(ℂ, F) f U) (hpc : IsPreconnected U) (hc : IsCompact U)
    (ho : IsOpen U) (ha : a ∈ U) (hb : b ∈ U) : f a = f b 
goal: f a = f b

full_name: MDifferentiable.isLocallyConstant
code: /-- A holomorphic function on a compact complex manifold is locally constant. -/
protected theorem isLocallyConstant {f : M → F} (hf : MDifferentiable I 𝓘(ℂ, F) f) :
    IsLocallyConstant f 
goal: IsLocallyConstant f

full_name: MDifferentiable.apply_eq_of_compactSpace
code: /-- A holomorphic function on a compact connected complex manifold is constant. -/
theorem apply_eq_of_compactSpace [PreconnectedSpace M] {f : M → F}
    (hf : MDifferentiable I 𝓘(ℂ, F) f) (a b : M) : f a = f b 
goal: f a = f b

full_name: MDifferentiable.exists_eq_const_of_compactSpace
code: /-- A holomorphic function on a compact connected complex manifold is the constant function `f ≡ v`,
for some value `v`. -/
theorem exists_eq_const_of_compactSpace [PreconnectedSpace M] {f : M → F}
    (hf : MDifferentiable I 𝓘(ℂ, F) f) : ∃ v : F, f = Function.const M v 
goal: ∃ v, f = Function.const M v