Theorems · Theorem · several complex variables

Complex.norm_eventually_eq_of_mdifferentiableAt_of_isLocalMax

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {F : Type u_2} [inst_2 : NormedAddCommGroup F]
  [inst_3 : NormedSpace ℂ F] {H : Type u_3} [inst_4 : TopologicalSpace H] {I : ModelWithCorners ℂ E H} [I.Boundaryless]
  {M : Type u_4} [inst_6 : TopologicalSpace M] [inst_7 : ChartedSpace H M] [IsManifold I 1 M] {f : M → F} {c : M},
  (∀ᶠ (z : M) in nhds c, MDiffAt f z) → IsLocalMax (norm ∘ f) c → ∀ᶠ (y : M) in nhds c, ‖f y‖ = ‖f c‖

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.

Defined in

Mathlib.Geometry.Manifold.Complex

Cited by

1 results in Mathlib

Foundations

Depth 289 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceModelWithCorners.BoundarylessTopologicalSpaceChartedSpaceIsManifold