By Luther Pfahler Eisenhart

Created in particular for graduate scholars by means of a number one author on arithmetic, this advent to the geometry of curves and surfaces concentrates on difficulties that scholars will locate such a lot priceless.

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**Extra resources for A Treatise on the Differential Geometry of Curves and Surfaces**

**Example text**

Then ı u is subharmonic in . uj /j 2N be a decreasing sequence of subharmonic functions in u WD lim uj is subharmonic in . "j / 2 RN be such that j 2N "j < C1. C P Then u WD j 2N "j uj is subharmonic in . z; x/ W X ! t. e. x 2 X , z 7! t. e. x 2 X . R Then z 7! x/ is subharmonic in . and g 2 L1 . / Proof. (1) is an immediate consequence of Jensen’s convexity inequality. (2) It is clear that u D inffuj I j 2 Ng is usc in . The submean value inequality is a conseqence of the monotone convergence theorem.

5. Let fj W R ! R be a sequence of convex functions which converge pointwise towards a function f W R ! R. fj / converges to f uniformly on each compact subset of R. 6. 1. Compute the Laplacian in polar coordinates in C. 2. jzj/, a smooth function in Œ0; RŒ. r/: r 3. Describe all harmonic radial functions in C. 4. 0; R/ iff function of t D log r in the interval 1; log RŒ. 7. Let h W R2 ! R be a harmonic function. x/j Ä C Œ1 C jjxjjd ; for all x 2 R2 : Show that h is a polynomial of degree at most d .

Show that any convex function f W R ! R that is bounded from above is constant. Use this to prove that any plurisubharmonic function ' W Cn ! R that is bounded from above is constant. 14. Let Cn be a domain. Show that ' W ! R is pluriharmonic if and only if it is locally the real part of a holomorphic function. 15. 1) Let ' W Cn ! R be a plurisubharmonic function. Rn /: 2) Let 'j be a sequence of plurisubharmonic functions in Cn such that 'j ! Cn /. Show that 'j jRn ! 16. z/ D j log jjzjj in Cn ?

### A Treatise on the Differential Geometry of Curves and Surfaces by Luther Pfahler Eisenhart

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