This result states interior regularity theorem for two-dimensional area-minimizing integral currents in r 3: given T be an area-minimizing integral 2-current in an open subset U R 3. For every point x 0 spt T spt T, there exist r>0, an integer q N, and a smooth embedded minimal surface.... It is useful in geometric measure theory and minimal surface regularity, where variational identities, curvature estimates, and compactness arguments control geometric objects.