# Chern-Osserman type equality for complete surfaces in R^n

## TL;DR

This paper establishes a Chern-Osserman type equality for complete surfaces in Euclidean space with finite second fundamental form norm and shows finite mean curvature norm implies quadratic area growth.

## Contribution

It introduces a new equality relating geometric quantities of surfaces and links finite mean curvature norm to quadratic area growth.

## Key findings

- Chern-Osserman type equality derived for surfaces with finite second fundamental form
- Finite L^2-norm of mean curvature implies at least quadratic area growth
- Monotonicity formula used to establish area growth behavior

## Abstract

We obtain a Chern-Osserman type equality of a complete properly immersed surface in Euclidean space, provided the L^2-norm of the second fundamental form is finite. Also, by using a monotonicity formula, we prove that if the L^2-norm of mean curvature of a noncompact surface is finite, then it has at least quadratic area growth.