By Demailly J.-P.

**Read or Download Complex analytic and differential geometry PDF**

**Similar differential geometry books**

**Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden**

This booklet illustrates the wide variety of Jerry Marsden’s mathematical legacy in parts of geometry, mechanics, and dynamics, from very natural arithmetic to very utilized, yet consistently with a geometrical viewpoint. every one contribution develops its fabric from the perspective of geometric mechanics starting on the very foundations, introducing readers to fashionable concerns through illustrations in quite a lot of subject matters.

**Geometry and Analysis on Manifolds: In Memory of Professor Shoshichi Kobayashi**

This quantity is devoted to the reminiscence of Shoshichi Kobayashi, and gathers contributions from special researchers engaged on themes as regards to his examine components. The e-book is equipped into 3 components, with the 1st half proposing an outline of Professor Shoshichi Kobayashi’s occupation. this can be by means of expository direction lectures (the moment half) on contemporary issues in extremal Kähler metrics and price distribution idea, with a view to be useful for graduate scholars in arithmetic drawn to new themes in advanced geometry and intricate research.

**Degenerate Complex Monge–Ampère Equations**

Advanced Monge–Ampère equations were essentially the most robust instruments in Kähler geometry due to the fact Aubin and Yau’s classical works, culminating in Yau’s option to the Calabi conjecture. A amazing software is the development of Kähler-Einstein metrics on a few compact Kähler manifolds. lately degenerate advanced Monge–Ampère equations were intensively studied, requiring extra complex instruments.

- Flow Lines and Algebraic Invariants in Contact Form Geometry
- Lectures On The Geometry Of Manifolds, 2nd Edition
- Gnomon: from pharaohs to fractals
- Surveys in Differential Geometry (Volume 12): Geometric Flows
- Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists

**Extra info for Complex analytic and differential geometry**

**Sample text**

G. ψ(ζ, z) = max{|ζ|2 + | Re z|2 , − log δΩ (ζ, z)}. There is slowly increasing sequence Cj → +∞ such that each function ψj = (Cj − ψ ⋆ ρ1/j )−1 is an “exhaustion” of a pseudoconvex open set Ωj ⊂⊂ Ω whose slices are convex tubes and such that d(Ωj , ∁Ω) > 2/j. Then 1 vj (ζ, z) = v ⋆ ρ1/j (ζ, z) + | Re z|2 + ψj (ζ, z) j is a decreasing sequence of plurisubharmonic functions on Ωj satisfying our previous conditions. As v = lim vj , we see that u = lim uj is plurisubharmonic. 6) Corollary. Let Ω ⊂ Cp × Cn be a pseudoconvex open set such that all slices Ωζ , ζ ∈ Cp , are convex tubes in Cn .

If R < 1, we get a contradiction as follows. Let ψ ∈ Psh(Ω) be an exhaustion function and K = h ∂D(1) × D(R) ⊂⊂ Ω, c = sup ψ. K As ψ ◦ h is plurisubharmonic on a neighborhood of D(1) × D(R), the maximum principle applied with respect to t implies ψ ◦ h(t, w) c on D(1) × D(R), hence h D(1) × D(R) ⊂ Ωc ⊂⊂ Ω and h D(1) × D(R + ε) ⊂ Ω for some ε > 0, a contradiction. d) =⇒ e). The function − log d(z, ∁Ω) is continuous on Ω and satisfies the mean value inequality because − log d(z, ∁Ω) = sup − log δΩ (z, ξ) .

Convexity Properties The close analogy of plurisubharmonicity with the concept of convexity strongly suggests that there are deeper connections between these notions. We describe here a few elementary facts illustrating this philosophy. B (Kiselman’s minimum principle). 13) Theorem. If Ω = ω + iω ′ where ω, ω ′ are open subsets of Rn , and if u(z) is a plurisubharmonic function on Ω that depends only on x = Re z, then ω ∋ x −→ u(x) is convex. Proof. This is clear when u ∈ C 2 (Ω, R), for ∂ 2 u/∂zj ∂z k = 41 ∂ 2 u/∂xj ∂xk .