We show that all meromorphic solutions of the stationary reduction of the real cubic Swift-Hohenberg equation are elliptic or degenerate elliptic. We then obtain them all explicitly by the subequation method, and one of them appears to be a new elliptic solution.1 The value a 0 here is different from that in [15]. In [15, p. 2040], a 0 = β 2 0 , where β 0 = √ 2/k 0 , and 4k 2 0 − 2k 0 − 3 = 0. The value a 0 stated in [15] is probably a typographical error.

For a noncompact complex hyperbolic space form of finite volume X = B n /Γ, we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification X of X similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity X − X called canonical radius of a cusp of Γ. The main result in the article is that there is a constant r * = r * (n) depending only on the dimension, so that if the canonical radii of all cusps of Γ are larger than r * , then there exist symmetric differentials of X vanishing at infinity. As a corollary, we show that the cotangent bundle T X is ample modulo the infinity if moreover the injectivity radius in the interior of X is larger than some constant d * = d * (n) which depends only on the dimension.

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Let $M$ be a Carathéodory hyperbolic complex manifold. We show that $M$ supports a real-analytic bounded strictly plurisubharmonic function. If $M$ is also complete Kähler, we show that $M$ admits the Bergman metric. When $M$ is strongly Carathéodory hyperbolic and is the universal covering of a quasi-projective manifold $X$, the Bergman metric can be estimated in terms of a Poincaré-type metric on $X$. It is also proved that any quasi-projective (resp. projective) subvariety of $X$ is of log-general type (resp. general type), a result consistent with a conjecture of Lang.