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The Diederich-Fornaess index has been introduced since 1977 to classify bounded pseudoconvex domains. In this article, we derive several intrinsic, geometric conditions on boundary of domains for arbitrary indexes. Many results, in the past, by various mathematicians estimated the index by assuming some properties of domains. Our motivation of this paper is, the other way around, to look for how the index effects properties and shapes of domains. Especially, we look for a necessary condition of all bounded pseudoconvex domains Ω ⊂ C 2 with the DiederichFornaess index 1. We also show that, when the Levi-flat set of ∂Ω is a closed Riemann surface, then the necessary condition can be simplified. IntroductionLet Ω be a bounded domain in C n with smooth boundary andThe Ω is said to be pseudoconvex if − log(−δ(z)) is plurisubharmonic in Ω. Note that − log(−δ(z)) blows up whenever z approaches the boundary ∂Ω. Indeed, all bounded pseudoconvex domains with C 2 boundary admit bounded (strictly) plurisubharmonic functions which vanishes on ∂Ω was shown by . They proved that any relatively compact pseudoconvex domain in Stein manifolds admits a defining function ρ such that −(−ρ) η is (strictly) plurisubharmonic in Ω for some η ∈ (0, 1]. The author also remark that the ρ may not be δ in general. For the pseudoconvex domain in complex manifolds, see Range [21].For the bounded pseudoconvex domain Ω with C 1 boundary, Kerzman-Rosay constructed a smooth (strictly) plurisubharmonic function in [17]. This function also vanishes on the boundary ∂Ω. Later, Demailly in [8] improved the result of Kerzman-Rosay to arbitrary bounded pseudoconvex domains in C n with Lipschitz boundary. He also showed his plurisubharmonic smooth function φ is bounded above and below by a multiple of − 1 log(−δ) near the boundary. In [14], Harrington found a new smooth (strictly) plurisubharmonic φ in bounded pseudoconvex domains in C n with Lipschitz boundary. In this paper, the φ is Hölder continuous on the boundary ∂Ω. In [13], he also obtained some results about pseudoconvex domain with Lipschitz boundary in CP n .If the boundary is smooth or at least C 2 , then the above result of Diederich-Fornaess is applicable. In particular, the fact that −(−ρ) η 0 is (strictly) plurisubharmonic will implies that −(−ρ) η is

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The Schwarz lemmas are well-known characterizations for holomorphic maps and we exhibit two examples of their applications. For a sequence family of biholomorphisms f j , it is useful to determine the location of f j (q) for a fixed point q in source manifolds (see Proposition 2.1). With it, we extend the Fornaess-Stout's theorem of (Fornaess and Stout, 1977) in monotone unions of balls to ellipsoids in Section 2. In Section 3, we discuss the curvature bounds of complete Kähler metric on ⋊ domains defined in (Liu, 2014) with an idea of (Yang, 1976).